Something that I don't think is sufficiently appreciated is that to obtain the kind of very rigorous immigration enforcement that people think they want would require steps that people don't actually want.
To take a silly example, if you tried to prevent illegal immigrants from mooching off the taxpayer by visiting the National Gallery of Art, you'd need to annoy every single citizen, tourist, and legal immigrant by checking papers at the door. It would be a huge pain in the butt and people would, legitimately, complain. More seriously, for all the talk of "build a way" and "secure the border" one of the biggest lapses in our border security regime is that any citizen of a Visa Waiver Program country can just hop on a plane and fly over here no questions asked. If you're Greek, tired of the 27 percent unemployment rate, and your uncle's a plumber in Chicago who's happy to help his family out by taking your on as an apprentice, then there's nothing standing in your way.
We could secure the border by eliminating the Visa Waiver Program but we didn't create the program in a fit of absent-mindedness. It's good for tourism and tourism-related industries as well as for multinational companies that folks can travel conveniently from Europe, Taiwan, Japan, and South Korea easily and conveniently. What's more, if we started requiring visas to enter the country some of those other countries might reciprocate which might be even more annoying. The traditional calculus has been that these are rich enough countries that we don't need to worry that large numbers of people want to come here and work off the books, but that's a question of trade-offs.
And you have the same thing on the employer side. There's a woman who cleans my house on alternate Wednesdays. I didn't ask to see a green card when I hired her, and more to the point I would have no idea how to recognize a fraudulent green card even if she'd shown me one. Tough employer sanctions sound good, but stories about middle class families and local small businesses getting hit with tough fines for being duped by fake documents sounds less good. Conservatives are usually well-aware of the fact that even well-intentioned regulations can have a lot of compliance costs for people and firms who aren't really doing anything wrong, but immigration enforcement is just a special case of the general principle. That's not a reason to have zero immigration enforcement any more than it's a reason to allow for unlimited air pollution. But enforcement isn't costless and doesn't just happen "at the border." We want people to be able to come to the United States to visit, and once they're here it's difficult to entirely prevent other people from giving them money in exchange for doing useful work.