This is a good aspect to use to make your character three dimensional. Everyone has fears even our brave heroes and heroines. As writers we want them to have fears because fears lead to internal and even external conflicts. Conflict is what drives your story forward. Without it the story will stall and die–not a pretty sight, especially if we have invested many hours in writing the story to that point. The plot comes from the characters and their conflicts and fears. These characters have to grow and change which comes from dealing with their conflicts and fears. This should be a gradual change. A sudden one won’t be believable.
Again anything is possible so long as it makes sense with everything else you have developed in the book. Fears stem from a character’s background (as does his goals and motivations). Without fears a character won’t seem human. Fears and conflicts are what can give your character flaws and vulnerability–make your reader root for him. A perfect character, as I said earlier, would be boring. There are a lot of fears to draw on: fear of intimacy, lack of control, exposure (ties in with secrets), being alone, making a fool of himself, commitment. I think you can get the picture. This will be the meat of your story–where the drama will come in.
In GOLD IN THE FIRE, my October 2004 Love Inspired, Joshua fears being hurt again because he was left at the altar. You see how I used something in his past to develop his fear. As I develop a character’s background, I keep that in mind. Fears don’t come out of the blue with no basis for them. The same with flaws. There should be a reason a flaw is there and there should be a flaw.