I've been climbing for over 10 years and have been an instructor for 5. When I first started climbing, the idea of falling kept me from getting half way up the wall. However, within a couple weeks of my first attempt, I was trusting my gear and my climbing partner. That trust is what allowed me to deal with my fear. I can hang from the side of a rock face with little to no fear. Put me on an extension ladder 20' up, however, and that fear is right back again.
If you want to get over you fear of heights for climbing, then you will want to do 2 things:
1) Support System: a climbing partner that you trust and is looking out for your interests will help push you beyond your comfort level but not to the point of panic.
2) Keep coming back: you won't learn to deal with the fear if you are only going in to climb once in a while. Coming in a couple times a week will allow you to remember how it was last time. You will also want to try to get a little higher each visit.
The above advice is more geared for teens and adults. It can help with kids, but sometimes it is something else that is stopping them. We had a 6yr old climber associate his fear with a specific hold. The first couple times he climbed, I am sure his fear is genuine. After that however, he would stop at that hold and come back down even though he wasn't showing any signs of fear up to that point. We set some new routes in the rock wall and coincidently removed that hold from the wall. The next time he came in, he climbed all the way to the top because he never encountered his hold. He has been able to be successful on the other walls now due to this breakthrough. If you have a child who always stops at the same hold, but appears to be comfortable up that point, let your rock wall staff know it may be time to redo that area of the wall.
I have been climbing for over 10 years, and an instructor for 5. When I first started in the sport, the idea of falling kept me from getting even half way up the wall. Climbing has helped me deal with that fear. I learned to trust my gear and my belayer. I very quickly was able to get to the top of a route. However, if you put me on an extension ladder, then I'm back to square one.
Two things you can do to get yourself comfortable with climbing:
1) Support system: you need someone that will climb with you that understands you are scared, will help push you just outside your comfort zone but not to the point of panic. If you don't have someone that will do this with you, ask your friendly rock wall staff member to help you out. Some of us have been in your position, but have been able to overcome.
2) "Caution" zone: Everyone has a traffic signal in their heads. When you are engaged in an activity that you don't even think about the risk w/o fear, you are green. Once you become scared but can still function, you are yellow or caution zone. If you are to the point of hysterics, and/or frozen panic, you're in the red. While climbing, push yourself into the yellow or caution zone and then stop. Don't let go of the wall, just stay there and think about everything that is in place keeping you safe. What will happen if you fall? If you want to come down then come down. Next time, however, try to get one hold higher on the wall.
These techniques will help you DEAL with your fear, but know that the fear doesn't really go away, you simply get more comfortable in the situation you are in.