Interestingly enough, I've gone through this. My wrist/hand pain began with computer usage. Specifically mouse clicking.
Smash never really gave me any problems until a week ago. I began working on doing Fox and Falco's waveshines as quickly as possible, and began having more intense pain in my wrist than I ever have before. I believe the direct cause was clicking the R button in so aggressively.
So, I did my homework and am now playing pain-free, aside from the soreness I'm still recovering from. I'll tell you what worked for me:
The MOST important change I've made is staying relaxed at all times. When you're playing, do NOT tense up your hands. Relax the shoulders and the neck as well - they are directly correlated with your wrists/hands.
To get a feel for it, do this: Stand up, put all of your weight on your feet, and completely relax your shoulders and neck. This should give you a feel for the difference in relaxed vs not-relaxed shoulders.
Also, do wrist massages. I ran across this video on YouTube
http://www.youtube.com/watch?v=SR3uqanXXxg and it seems to make my wrists feel a lot better.
Another thing I changed: I stopped wavedashing. I stopped playing Fox and Falco and went to slower characters that don't really require wavedashing: Peach, Link, and Marth primarily. (Marth's dashdance is so good that you rarely NEED to wavedash) Just do this until your hands and wrists heal up. It's possible to play extremely well without wavedashing. L-cancelling is still a must, but it requires little to no strain on the hands and wrists because you don't have to fully click in the shoulder button. You can just tap it.
Next: Posture. When you're sitting down and playing, keep your shoulders relaxed like I said, and try to keep a straight back. It takes a little effort if you're used to slouching, but it's worth it IMO because I'll do anything to have no pain and continue to play my favorite game. I've been doing this for days now, and it's gotten easier and easier to sit in an upright position for long periods of time.
RELAX THE SHOULDERS! I feel like I can't stress this enough. Not only has it reduced my pain, but I actually play a LOT better now. I never tense up, which means my fingers are always loose and able to move quicker, and this has also resulted in me being calm no matter what's happening in the match. The benefit of this: I don't screw up when I'm down on stocks, my precision and reaction have gone up in consistency SO MUCH. It's ridiculous.
TL;DR:
Sit upright, RELAX SHOULDERS, keep hands loose and don't tense them up. Just stay calm, relaxed and work on timing. You don't have to mash buttons or grip the controller like you're actually going to make your character survive longer. Everything has a timing.
Oh, and stretch and massage daily. It helps a lot.