This is really neither here nor there as far as the current politics of immigration go, but it is worth taking the opportunity now and again to point out that securing the southern border is a pretty dumb approach to immigration control. It’s extremely hard to do it with any degree of efficacy in a way that doesn’t seriously impede commerce and tourism. Meanwhile, even when it is done effectively, it still leaves all kinds of other routes — overstaying visas, coming on a boat, etc. — open. Last, if someone happens to be physically inside the United States for some period of time — jumping back-and-forth over the board for fun, or heading into some border town for the afternoon to buy something — it’s really not what we’re worried about.
Conversely, if you can make it really difficult for visa-less person to get a job, rent an apartment, etc., then this will dramatically curb illegal immigration while simultaneously allowing the government to not spend a huge amount of effort on hassling legitimate border-crossers.
To do that, all you need to do is establish a hefty incentive for illegal immigrants to rat out people who illegally employ them. Mark Kleiman has proposed a “poetic justice” version of this where an illegal who rats his employer out gets a green card in exchange. More prosaically, a ratter out could get a one-way ticket back to his home country plus a big fat check financed through employer fines. An enforcement system like this would be cheap to administer since you mostly wouldn’t need to administer it at all — illegal immigrants looking for a bonus, and potential employers of illegal immigrants afraid of being caught in a sting, would do the vast majority of the work.