Should I freeze gyoza before or after cooking? I'm about to make a big batch of gyoza and I don't want to eat them all now. I was thinking about freezing them and I want to know which is best from a food-safety and a quality perspective. 

If I freeze them after cooking, is the crispy side going to become soggy after cooking? If I freeze them before cooking, are they going to get soggy whilst the meat inside defrosts? Is it safe to defrost them with raw meat inside?
Either would be safe.

You will get better quality by freezing them uncooked.   They should not get soggy due to the freezing, as neither the meat filling, nor the wrappers, are particularly subject to ice crystal damage.

Gyoza are very small, so you should not need to thaw them:  just begin the steaming phase right out of the freezer, and they will thaw while cooking.   If you do wish to thaw them, do it in the refrigerator and they will be perfectly safe, despite the raw filling.

You are correct that if you freeze them after cooking, you will loose the crispy bottoms--this would be true if you save them as left overs in the refrigerator as well.