Oloenneker has stated a couple of things that I would like to address.
1. that liberals' position is to go after the employers. I don't think that liberals have the corner on that market by any stretch. Economic liberals tend to favor open borders in the name of free trade(as opposed to managed trade). Then you have the socially liberal. I would venture a guess that they are in part just as responsible for hiring these illegals as the so-called conservatives. Neither side will come out ahead on that, just a bunch of finger-pointing.
So going after the employers can be considered a centrist position without any of us becoming rabid nationalists. For example, I am part Finnish by blood, 100% U.S. citizen by birth and yet I would advocate prosecuting the employers, leaving very little deporting of individuals as any "solution" to this issue.
2. Benefits are oriented toward lower income brackets and this does include many migrant workers. Until I made more than $48K/yr., my wife was getting offers from the state for WIC. We don't use it, but their offer still stands. Now most years I don't even qualify anymore.
I can tell you that very often being behind the migrant workers in line at Walmart the mama will pay for food with foodstamps and the papa will have a bankroll in his pocket that would make you blush. This in and of itself doesn't prove that they are illegal (until you hear their English and see their tan)or that they are a problem. Just that the benefits are geared toward a certain financial sector, and that for the time being, those folks fit into the guidelines. It may seem anecdotal, but I disagree with it being a misnomer. It's not a misnomer. Look around.