Should I freeze gyoza before or after cooking?
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.