Simple background check like they normally do with any other employee: photo ID (drivers license), birth certificate (which all natural U.S. citizens have) or naturalization papers / passport/visas/green-cards for legal immigrants.
It's a routine part of normal hiring practices.
And, just how are these documents supposed to be verified? Or are the businesses just supposed to take the applicants' word for their authenticity?
Many large coporations already get inspected periodically to see if they are within the law. But when it comes to mom and pop operations it is much more difficult to prove. Especially if the Dept of Labor is issuing out labor certs like candy.