No wall is needed... Just make it so that employers cannot legally employ those not authorized to work in the country --and fine with an amount which will deter people/companies.
At least that's what I would do if I needed to control unauthorized workers for any place which wanted to do so.
Well, it's a good thing you don't. Fines that high would allow some random mid-level government bureaucrat to effectively shut down any business that crossed him or her.
No creditor is going to do business with an entity that could cease to exist tomorrow. Even an accusation of hiring illegal immigrants would effectively be a corporate death penalty, as the business would find itself quickly frozen out of its relationships with customers, creditors and suppliers. In theory this leads to businesses scrupulously following the rules, avoiding even the appearance of violations. In practice, though, rules are complicated and ambiguous, time is finite, competency is finite, and even the most well intentioned business can hire someone without the correct paperwork. Or hire someone, and then have their paperwork expire. Or miss a required check on paperwork. Or something else entirely. Are you going to punish a business with death for making a simple error in filing horrendously complicated immigration paperwork? If so, do you also think that we should have police snipers ready to kill anyone going even one mile over the speed limit? After all, speeding kills many more people than illegal immigration every year.
If you can't be bothered to get documents from an employee and e-verify their employment eligilbity in 5 minutes on a web app, you shouldn't be permitted to run your business.
Because something does not have absolute accuracy does not mean you do not do it. Pass, fail, and refer are perfectly acceptable, with a legitimate and efficient appeals process if you truly are eligible to work and are denied through the verification process.
It already is illegal to employ illegal immigrants, and in fact most businesses that do, are also committing tax fraud, which can be punishable with jail time.
At least that's what I would do if I needed to control unauthorized workers for any place which wanted to do so.