Punish those that will hire illegals then... shut them down. Right?
Absolutely right!
The problem is the ready availability of fake ID. Anything from state driver licenses, birth and baptism certificates, to military separation papers can be bought in Tijuana or from dealers who the employers of illegals direct them to. The quality ranges from poor to excellent but quality doesn't matter as long as the illegal has the documents when he's working in case the INS pulls a raid.
This is the reason Congress has resisted passing a law requiring a biometric ID card which is registered and addressable (like credit cards) via swipe terminals carried by INS agents. These addressable cards would be verifiable by fingerprint and/or iris reading. It would take six months to complete a program requiring all citizens to have a card. The cost could be offset by charging those above a given income level $10.
Such a program, backed by a significant penalty for violations, would put an end to the hiring of illegals, the vast majority of whom would be forced to voluntarily go back to Mexico and Central America. But our corrupted, corporatist Congress won't pass such a law. In fact some of them discourage it by saying such a requirement is "un-American."