How many Americans dream of doing farm work? We have some of the best farmland in the world, but our workforce has mainly moved up the tech chain to more skilled and hence more valuable labor. It is a contradiction.
I worked on a farm in CA as a seasonal worker and I, along with a number of my fellow laborers, were paid below CA minimum wage, and I was actually authorized to work in the US. I don't know how widespread this is but I got the impression it wasn't exactly rare.
From comparing experiences with one of my closest friends who is the son of Mexican migrant farm workers in the Central Valley, we have concluded that some farm operators specifically seek out illegal workers, not so much because the hourly wage differential is so great, but because of added savings like insurance and not having to deal with OSHA violations. Farm work an be pretty injury inducing and having a group of workers who are too scared/desperate to report worker safety violations to OSHA can be a big benefit.
For the sake of not generalizing too much, I also know someone who runs a small farm and takes things like paying legal wages and overall worker safety and well being extremely seriously. And again, this is all anecdotal.
When I propose ending the work visa program to 'conservatives', I am laughed at. Maybe there's a better solution, but it's at least a more direct solution to their stated woes than I've heard. Not to put liberals on a pedestal either though - the left in this country basically ignores AG labor entirely.