There's no magic bullet, and, while coaches and parents can try to provide kids with tools to manage fear, ultimately it is the child her/himself who has to do it. The boys are just starting to do scary skills, and their coach tells them, "don't let your fear interfere with what you want to do," but ultimately, unless they really, really want it, they won't be able to work through the fears. My answer is provide the tools, listen when the gymnast initiates the conversation with you, don't let her/him have a pass by saying the coaches are mean by pressing her/him to confront the scary skill, and then stay out of it. They have to figure it out. Or not.
I write this as a parent whose daughter was making great progress on beam -- as she was wrapping up her L6 season in spring 2012, she got her handstand-BHS on the high beam. Did 100 of them, duly logged on the white board. And then she lost it.
For nearly two solid years.
She cried at night after practice. She had trouble sleeping. We did some Doc Ali. Her coaches were encouraging. Her coaches were tough. At the beginning, I initiated conversations with her and tried to give her strategies. She did the skill at home on her low beam. She did endless open gyms. I offered to get her privates. I stopped talking about beam at all unless she initiated the conversation. She still initiated conversations about how miserable she was. She dialed back and did every drill imaginable and did probably thousands of repetitions of the skill on various low beams, pads, beams with resis, etc., etc. She was thoroughly and completely unhappy but did not want to quit.
She repeated L7 and competed a walkover walkover all year while watching her teammates upgrading. Judges are picky about that connection. Mostly she got credit. Sometimes she didn't. Not getting credit made her angry, but she knew the only solution was an upgrade that she couldn't do. She did pretty well in meets by the end of the year and made great progress on all the other events. And maybe that's what convinced her that she wanted -- really wanted -- to be a L8 next year. She wanted it a lot. She wanted it badly, in a way that she hasn't wanted anything in gymnastics for a long time.
The HS-BHS is now back, this time, I think, for good.