Employers have to do that now. When a person is employed they have give their social security number for tax purposes. Employers also have to verify that the person is of legal age to work in that state.
If employers can't be blamed, then the guys issuing the phoney SS#s have to be made an example of. Make it a HIGH priority to catch the guys giving out these SS#s and put them on the news doing their perp walk to jail for 10 years.