Earlier quoted context omitted.
Punish employers with heavy fines for e-verify violations. If no one hires you, no reason to stick around. This is what Japan does, Korea, Taiwan, etc. The US is way lenient on illegal entry and accommodating to irregular workers. In many countries the only way you get around that is if you work with the local mobsters --so only highly risk taking indivs take that opportunity.
IIRC employers avoid this liability by executing the fraud themselves. There's no e-verify violation because they use legitimate SSNs from a living person, fraudulently. The workers aren't the problem, the employers are. Legislators and govt executives look the other way in order to preserve competitive agriculture.
IE, if you get "tricked" via a stolen SSN, then the business gets punished regardless.