I don't think anyone supports illegal immigration. That would an indefensible position.
Illegal immigrants have done the dirty work for generations. Legal immigrants did many of them also because showing up at a port of entry and not having any communicable disease was all you needed to gain legal entry. That's how it was done in the past. The immigration laws have changed. The US, not unlike any other developed nation, allows only the cute and fuzzy immigrants now - the educated and rich ones - to immigrate. Unskilled laborers cannot immigrate now, so the low-skill positions are filled by many illegals. Americans are not lining up to fill out applications for those positions. If they were, they would be working in them.
Prices didn't go down with lower wages; executive, owner and investor compensation went up. Prices only go down when forced down by competition, not by reducing costs. Prices go up whenever any reason that sounds plausible can be thought up.
As usual, lot's of talk about problems, but never any action. One man's solution is another man's economic loss. It would help Arizona (the highest violent crime rate in the US, higher than South Africa and other 3rd-world countries) if the US enforced the federal immigration laws with employers. The federal government is doing things it shouldn't be doing and few of the things it should be doing.