As long as there is economic inequity (which is to say, forever) there will be illegal immigration.
The US, as mentioned, is not the only country having this experience and struggling with what to do about it. You may be surprised to know that Mexico itself has an illegal immigration problem in the South. Hondurans often illegally work in the Dominican Republic. There are what are commonly termed Push/Pull factors that drive millions of people all over the world to leave their homelands and go to other countries to work. In Mexico - the country with which I am most familiar - there is the push factor of no jobs/horrible wages/malnutrition and the pull factor of the huge economic engine that is the US.
Complicate that with the fact that Mexico depends on money sent back from its citizens abroad. Money coming back from the US is the third largest income Mexico makes, after oil and tourism.
In the US, the employer is responsible for verifying eligibility to work, but the way the law is written, as long as the employer makes a good faith effort he’s pretty much in the clear. When INS started Operation Vanguard in Nebraska, it hit the meat packing plants very hard. Pressure from companies like IBP and the politicians stopped it. Meat packing in the Midwest relies on undocumented immigrants. They don’t want the law enforced. INS is notoriously underfunded and under-resourced. All those people who want to enforce the law have to accept higher prices, yes, but the US Congress also turns a blind eye in its funding. Where do you think the pressure comes from in that case?
And when we’re talking about illegal immigration, it’s assumed we’re talking about dark skinned folk from Latin America. But what about all the visa overstays (which can be tourism, H1-B’s, higher education)? There are many, many of those, as well, but I don’t think those are the people the close-the-border folks are talking about.
I don’t see/hear anyone all freaked out about the influx of the Russians/Poles/Serbs, etc. Is that because they’re white?
Lastly, I’ve met/known quite a few people who have come to the US from Mexico to work. None of them really want to stay here. Most would happily go back to Mexico if they could take care of their families there. (And as a side note to those of you who worked hard jobs as teens: remember, the people we’re talking about have others than just themselves to provide for. It’s not really the same thing.)