Well, we do have a nod-and-wink economy, and that’s a genuine problem. Huge sectors of our economy rely heavily on undocumented labor precisely because they’re undocumented. Undocumented means various legal protections afforded to documented immigrants are more flexible–wages, leave, insurance, workplace safety. Do they report abuses? Pfft, no. Immigrants come here undocumented because there’s a demand for exactly that. If they had legal status, the under-the-table economy wouldn’t want them nearly as much.
Like the drug war, the problem needs to be addressed at the demand side. It’s not a choice between undocumented on one side and documented labor on the other. It’s a choice between livable wages and decent workplaces on one side and cheap food and labor on the other. We’ve chosen the latter, via a two-tier economy, one for us, one for them.