I've gotten skier's thumb twice in my right hand. The first time I went to this hand physical therapy doctor in Park City who made me a custom hand brace. I've seen pictures on here of the same thing before. I wore that for six weeks after each time, prevents it from moving so it can heal.
Really all you can do is let go of your poles as you fall and don't try to use your hands to break your fall. Each time I've done it I've held on to my pole, in which case it doesn't really matter how you use your strap (I don't use it, btw).