What is the point of debating someone if giving them arguments won't necessarily change their minds? That's what goes through the minds of many people when it comes to doing apologetics. Sometimes people reject the gospel because they have intellectual barriers to believing in Christ. That's where Christian Apologetics comes in, to tear down those barriers. Although often times people have more than just intellectual barriers, they also have emotional and/or moral barriers to becoming Christians (Romans 1:18, John 3:19-20). Even still, I don't even intend on convincing someone of my position. I've learned that rarely ever in a debate do you end up convincing the opposing side that you're right. However, what I do, I give arguments for Christianity and let The Holy Spirit move. If nothing else, I may be able to plant a seed in their heart that, through coming into contact with other Christian Apologists and witnesses, may have that seed of faith watered over time until they make that decision.
So, I basically just do apologetics, arguing as mightily as I can, and if they're convinced, I'll rejoice, but if not, then maybe they will be at a later date. Sometimes The Holy Spirit uses various witnesses to bring someone to Himself and also, sometimes getting someone to come to Him is a process that takes place over time.
When I do apologetics, I ask God to do His part by working on their heart and mind, and also to work on MY mind, so that I can successfully defend the faith and not get stuck, not knowing how to respond.
I don't agree that "apologetics is more designed for the believer" as some people sometimes say. I think it's definitely for the non-believer as well as the believer. Many non-believers have come to Christ through apologetics such as C.S Lewis, Lee Strobel, J. Warner Wallace, The skeptic Frank Morrison was converted while attempting to write a book refuting the evidence for the resurrection of Christ. Augustine tells in his confessions how he was led toward Christianity by hearing a Christian debate an unbeliever. By the way, this is another reason we should debate unbelievers. Even though the person we're debating may not be convinced, someone who's listening nearby may be (Lee Strobel calls this "Ricochet Evangelism").
Mark Mittelburg tells of another example in his book "The Questions Christians Hope No One Will Ask". He said that a Christian debated this atheist every day for a couple of years, soon the atheist began to question his worldview and eventually ended up abandoning it for the Christian faith. Hugh Ross, in one of the Reasons To Believe letters I get told a story of how an atheist came Christ after numerous, numerous discussions with him.
And again, I don't expect to convince anyone ( In fact, I expect my opponent not to be convinced) but I don't think it follows that we shouldn't dialogue with them because of that. I especially don't expect them to be convinced if this is the first time I've ever talked to them or the first time they've ever heard a Christian give intelligent responses to their objections and arguments, many Christians aren't prepared to defend their faith so some atheists assume there's no good reason to believe that The Bible is true and so when they come across a Christian who CAN give good reasons to believe, they're quite surprised and that discussion may be the first step in their spiritual journey to finding the answers and reasons they seek. Again, sometimes it takes many, many debates before a person changes their mind. Some people never will change their mind, they will go to their grave denying the evidence and rejecting Christ, but that should not deter us from engaging in meaningful discussions. As Christian Philosopher and apologist William Lane Craig said in his book "On Guard" "You can't expect them to just roll over and play dead..." after just one discussion. It might take a lot of discussions over an extended period of time, like with the atheist I mentioned in Mark Mittelburg's example. And it might also be that the discussion planted a seed in their heart that various Christians overtime will water and which will eventually blossom into a flower of faith. So therefore, I don't back down from a discussion with the mindset "This is a waste of time. he's not going to be convinced anyway. What's the point?"
Moreover, another reason for debating someone is that it can reassure you if the position you're defending is truly correct or not. As the old saying goes "beware the sound of one hand clapping" and as The Bible says "the first to plead his case seems right until another comes along and examines him." - Proverbs 18:17
I can tell you from my experience that I am far more convinced of the arguments in favor of Christianity when I actually use them in discussions and find how mightily my opponent struggles in trying to refute the premises of the argument. The more people I use that specific argument on and the more failures I see in trying to refute it, the more convinced I am that the conclusion is correct. When reading about or hearing the argument in isolation, it's easy to think "This sounds sensible, but what would the opposition say about it?"
I would like too add that even the Apostle Paul engaged in debates. He debated both Jewish leaders and Greek philosophers (Acts 17). Never in his epistles did The Holy Spirit inspire him to write that we shouldn't debate those who disbelieve The Bible but rather The Holy Spirit inspired him to write just the opposite
"We demolish arguments and every pretension that sets itself up against the knowledge of God, and we take captive every thought to make it obedient to Christ." - 2 Corinthians 10:5