Most illegal immigrants get a social security number through underground document-makers. They give that number to the employer (who knows it's fake, but acts otherwise).
I've befriended many Mexican immigrants, illegal and legal, and have even gone to a house that makes these documents. (It was surprisingly low-cost. )
Now this doesn't speak to your other point, about whether a farmhand job is livable for anybody. But my main point is: it's probably a legit job that withholds taxes and everything.
There is a large set of jobs that has a high-turnover rate for everybody who works in them. Overnight cleaning crews in offices, table bussers, farmhands... This turnover rate is high for everybody, legal and illegal, who works there. My guess is that the employers assume anybody who works there will leave soon, and it's not worth thoroughly vetting whether the ss# belongs to a dead guy.
As for my opinion on all this... I'm still working it out. I'm also biased, have lots of Mexican family on this of the border, and am admittedly sentimental about it. Ideally we can first focus on making any job livable, and THEN worry about who gets to have it.