Abraham Lincoln, the 16th President of the United States, once said "My great concern is not whether you have failed, but whether you are content with your failure."
The best way to learn not to fail IS to fail. That might sound a little odd, but take a moment and think about it. When you fail at something and you aren't content with that failure, you do something about it. You learn how to get back up. You learn that failure is something we all do from time to time, but we can overcome that - we can get back up and do better.
Knowing that you can get back up, helps with your fears of failure.