There are badges which you can receive for voting on questions/answers (Supporter, Civic Duty and Electorate) which are not hard to get when you are patient, because they do not require outside action (except that others have to post questions/answers). How about creating a similar set of badges which someone could receive for a total number of posted answers (for example with non-negative or positive score). I understand that answering the question isn't a hard action (voting is even easier), so the limits to acquire these badges would be probably around several hundreds or thousands.
I would not want to see people gaming to try and get this badge. There are already badges for posting good answers. I think this is preferable, since it rewards quality, not quantity.