Yes, forcing private citizens to do the work the federal government is supposed to do under threat of civil and criminal penalties is always the best way to go. This allows the fed to preserve precious resources to deal with pressing needs like free community college and partisan IRS auditing.
You act like these employers are so innocent. You do know employers advertise jobs in Mexico right?
It's one thing to fine/arrest someone for deliberately hiring an illegal laborer. It's completely another to require some uncertain degree of investigation and diligence, particularly when the fed is totally complicit in the [legal] immigration process.
It shouldn't be that hard to implement a nationally-mandated e-verify system where employers have to register their employees ssns, names, and birthdays. With that data, it would not be difficult for the Feds to ferret out stolen identities, and it also wouldn't be onerous on the employer.
Unfortunately, this sort of common sense legislation strengthening enforcement cannot even pass a republican congress because half of them are pro-amnesty.