Nope. Not if you set up the legal burdens correctly. The problem with the current system is that employers have the easy "out" from legal liability by claiming they relied on forged documents. Close that loophole with a better I.D., limit by statute the appeal process, and you can greatly increase enforcement.
The trick, though, is that you've got to give them an incentive to signup for the I.D.'s in the first place.
The incentive is to require the ID to transfer their money to Mexico. Once we cut down the flow to Mexico, the Mexican govt will feel the pressure and stop encouraging the immigration.