Fines and/or imprisonment for employers who KNOWINGLY hire illegals would get the job done. However, if an illegal has a phony but realistic Social Security card, it would be unfair to punish the employer. There was talk of creating a system whereby the employer could merely type in the social security number presented by the prospective employee, and a database would verify whether or not it was a valid number, without revealing the identity of the holder. I suppose they could give the year of birth to cut down on fakery without compromising identities.
The current system notifies employers when their workers are using false SSNs. Apparently there is no follow up.