First question is whether they want to copy both sides or just the front side. Since the SSN is on the back I'm not sure what the problem would be with them copying the front only.
If it is a hotel or other business that insists on copying both sides, you could always tell them you will take your business elsewhere.
As for DMV's, check with your state but I think most of them accept other forms of identification such as a DL from another state, birth certificate or passport.