First, I think it's helpful to clarify what a person really means when they identify themselves as "agnostic." In my experience, agnostics fall into one of three categories. Some agnostics are agnostic because they are unbelievers and they know it. They are being intellectually honest when they admit they don't believe in God, don't know Him, and therefore don't give any credence to the Bible which professes to be the Word of God. Others agnostics are spiritually lazy — they don't want to bother themselves to think about spiritual things or eternal consequences, and calling themselves "agnostic" throws up a sort of philosophical smokescreen they can take shelter behind that holds all perceived proselytization attempts at bay. This kind of agnostic essentially wants to be left alone, and embracing nominal agnosticism largely accomplishes this. Because it's impossible to empirically prove the existence of God, they will use this argument to deflect your evangelical advances; whatever evidence you provide, Scripture you reference, or standpoint you take as a Christian is skeptically viewed as being rooted in your personal, biased conviction of faith, and consequently, it can be rejected at will simply by citing their own lack of conviction. The last category of agnostics tend to be particularly quarrelsome; though they may appreciate and enjoy engaging in conversations about faith and beliefs, they are ambivalent and hesitant to commit themselves to any personal conviction about anything, preferring to limit the discussion to the theoretical and conjectural realm of philosophy.
In one sense, none of these, or any, classifications matter. Whatever labels we project onto people, the Bible says that all who reject God are unbelievers and, unless and until they are spiritually convicted and regenerated by the Holy Spirit, are destined for eternal judgment (Revelation 21:8). On the surface, there is a surplus of different reasons why a person will choose to reject God, but ultimately the root cause of human disobedience and rebellion is always the same: "But every man is tempted when he is carried away and enticed by his own lust. Then when lust has conceived, it gives birth to sin; and when sin is full-grown, it brings forth death" (James 1:14-15). The remedy is also always the same: submit yourself in faith to the Lord Jesus Christ and you will be saved from your sin and its deadly consequences (John 3:16; 5:24; 14:6; Acts 10:34-43; 1 John 5:20).
Romans 10:17 says that "faith comes from hearing, and hearing comes by the word of God." Since the Word of God is so essential to the sparking of faith, there is no way to effectively witness to someone apart from sharing the Word of God with them. God's Word has salvific power that no human argument for God's existence, as logical and well-reasoned as it may be, can ever have. But because both agnostics and atheists refute not only who God is and what He says, but
that God is, you have to start at the very beginning, at an earlier place than you would with a person who believes in God in some form or another. Apologetics can be a useful tool to communicate with, and begin to witness to, a nonbeliever who doubts or denies that God even exists. It is entirely possible to persuasively reason for the general existence of God, even to the personal God of the Christian Bible, as long as the person you're reasoning with is a reasonable person.
This is possible because God has deliberately left His fingerprints all over our universe. Romans 1:19-20 says, "...what may be known about God is plain because God has made it plain. For since the creation of the world, God's invisible qualities — His eternal power and divine nature — have been clearly seen, being understood from what has been made, so that people are without excuse." In addition to the external world, God has put the knowledge of Himself into every human being. The Bible says that all mankind, agnostic included, has an innate knowledge of God (Romans 1:21-28). But because of a sinful desire to do whatever we please without the restraint of acknowledging that God will one day hold each of us accountable, human beings have suppressed this instinctive awareness and have exchanged the truth for an assortment of more palatable lies (Romans 1:25), gorging themselves on the sin and depravity that leads to death (Romans 1:24-32). This knowledge should inspire us as Christians to desire to intervene and lovingly confront unbelievers with the truth that is able to save (James 1:21).
There is a natural, common sense progression in apologetic reasoning. To know God, one must start out by knowing that there is a God to know (Hebrews 11:6). Recognizing who He is and understanding what He requires of us both emerge out of that first step. The agnostic's problem is that he is unable to have a relationship with God since he cannot know for sure that there is a God to have a relationship with. With such an individual, scriptural proofs or doctrine will most likely not be well-received initially since God's Word probably won't carry much weight with someone who doesn't believe in God. This makes your job more challenging, but, thanks to God leaving clear evidence of Himself within us and within our cosmos, it can still be done. To do it, you need to reasonably establish that God exists, and you need to do it using logic, which is the only currency a true agnostic or atheist will deal in.
The existence of God can be reasonably argued from the principle of
cause and effect, which essentially states that no secondary effect can ever be produced without a primary cause. Evolutionists postulate that all of life came from a giant explosion or a puddle of primordial ooze, but where did the explosion-in-nothing or the puddle-in-nothing come from? Nothing can never produce something; it's an unreasonable and illogical impossibility to expect that it would or could — so eventually, as you go farther and farther back, you get to where there has to be something that preexisted everything, in order for there to ever have been anything.
The Bible uses the principle of cause and effect to make this exact same argument. Hebrews 3:4 says, "For every house is built by someone, but God is the builder of everything." If our universe is so complex and intricate in its structure, order, symmetry, and function that our best scientists don't even fully understand it, what are the chances that it all just came together in such an intricately ordered way? The Bible compares it to the possibility that a variety of scrap materials could randomly land in such a way as to result in a fully constructed, fully plumbed house. When you see a house, you know that someone built it even if you didn't witness them building it because of the apparent order and design and construction that went into it. It would be infinitely more ludicrous to look at the entire orderly universe and conclude it didn't have an architect and builder behind it than it would be to see a fully constructed house and conclude that it just happened to come together that way by a magical mixture of accident and time.
The law of cause and effect naturally flows right into the argument, not just for design, but for
intelligent design. When we're talking about us human beings, we're not just talking about a material house of sticks and stones and mortar. Every human being is composed of immaterial elements you could never get from purely material building blocks. How would you expect intelligence to arise from non-intelligence? How would you expect to get moral capacity, extremely complex sensory abilities, and mental capabilities like thinking and reasoning from primordial ooze? You wouldn't. You'd only get those highly complex, personal capacities from a highly complex, personal Source-Origin. Psalm 94:8-10 says, "Take notice, you senseless ones among the people; you fools, when will you become wise? Does He who fashioned the ear not hear? Does He who formed the eye not see? Does He who disciplines nations not punish? Does He who teaches mankind lack knowledge?" We live and have senses and a capacity to learn and think and act morally because God who sees and hears and thinks and reasons and judges designed us in His own image. We didn't accidentally spring from an impossible puddle of ooze, we were the deliberate, intentional creation of a Being who endows to us some of the abilities He Himself has and who placed us in a universe that irrefutably testifies to His existence (Romans 1).
These two progressive arguments — the law of cause and effect and the case for intelligent design — are extrabiblical, apologetic arguments that provide a starting place to reason with people who doubt or deny the existence of God. Of course, convincing an agnostic that God exists is just a first, tottering step towards the Gospel, but it is a monumental step for an agnostic or atheist as it opens the door to begin discussing who this God who exists, is.
At this point, evangelism will take a more regular and familiar shape, and the Gospel can be brought in to comment on the self-revelation of God in the Bible, as well as to offer cogent explanations for sin, evil, fallenness, guilt, and death, and finally to provide the ultimate resolution to the human dilemma in the form of God's gracious provision of forgiveness, reconciliation, and eternal life through Jesus Christ. A full (but simple) gospel presentation always includes these three facts in some form or fashion: 1) God is holy (1 Samuel 2:2; Isaiah 40:25; 57:15; Hosea 11:9; John 17:11; Revelation 4:8). 2) We are sinners, deserving and destined for death in hell (Genesis 6:5; Psalm 51:5; Matthew 15:19; Romans 3:28; Romans 5:12; Romans 8:7-8; James 1:14-15). 3) Christ died so that he could take the place of sinners who put their trust in Him (Romans 5:6; Ephesians 1:4-7; 1 Timothy 2:5; Titus 3:4-7; Hebrews 9:14, 28).
I would strongly recommend that you investigate the subject of apologetics in more detail than I can provide here within the constraints of a short article. Apologetics is a wide realm but there are many resources that can help you navigate and prepare yourself to use it in doing spiritual battle. See
CARM's apologetics section for a great introduction.
Additionally, I recommend picking up a copy of CS Lewis' apologetic work
Mere Christianity and Lee Strobel's investigative
The Case for a Creator and
The Case for Christ. These are all three highly useful tools to solidify why Christians believe what we believe, to refute nonsense propaganda that Christianity is an irrational belief system that is stuck in the Middle Ages, and to prepare you for confronting atheist, agnostic, and other unbelievers with logical, well-reasoned arguments. Lewis and Strobel, both former atheists, wrote these books to defend Christianity from its detractors and to provide consistent, progressive, and reasoned arguments for its validity and credibility.
Finally, as a general witnessing practice, it helps take a lot of pressure off us to remember that success in witnessing and ministry is not our burden to bear; it is only appointed to us to speak the truth in love (Ephesians 4:14-15), and to leave the outcome to God, trusting that He will work His will in His way and in His time. As industrious at planting, and as nurturing at tending and watering as we can be, it's still God alone who is able to make seed take root in soil and grow (1 Corinthians 3:7). I have watched seeds in optimum, greenhouse conditions remain dormant and come to absolutely nothing, and I have seen seed in the poorest, most impoverished soil take root and flourish, and I have learned that God's hand is behind both the failed seed and the mature plant. Our job as Christians is simply to do our best sowing and watering, and leave the rest to God who is the best and wisest gardener.