Here's a starting point:
Firmly block illegals, allow documented migrants in to fill in during harvests, etc, let the wages settle to an arm's length negotiated natural level*, get them back home when done.
We have a program for doing just that. Many (dare I say "most"?) of these (and many other) illegals aren't interested in participating because of that whole "get them back home when done" part.
Maybe I need more coffee, but that business of overwhelming punishment seems pretty unwise and fraught with unintended consequences all by itself.
I have to disagree. There's a reason why the usual suspects prefer to call illegal aliens "undocumented immigrants." Two reasons, in fact, both contained in the simple phrase. One is that the illegals are immigrants, they are not migrants. They don't want to go back home seasonally, and they don't. They want to stay here, taking jobs away from people who are here legally, and siphoning money out of the U.S. economy to send back to their native country. Second, anyone who is here legally has a document to prove that they're here legally. Birth certificate, passport, green card ... something. For every job I've taken or been in the final cut for over the past fifteen (maybe more) years, I have either had to show proof that I'm legally entitled to work in the U.S., or I've been alerted that I will have to produce such documentation if hired.
In fact, that's the law. I don't see how any employer today can possibly claim they "unknowingly" hired an illegal, unless the said illegal had a very well executed counterfeit birth certificate, passport, or green card. Consequently, I'm fully in favor of punishing employers who hire illegals. Those who do often pay them under the table, at less than minimum wage, so they not only prevent legal citizens and legal residents from finding work, they also drive down the prevailing wage for the area.
Maybe it's just sour grapes on my part. When I married my late wife, we jumped through all the hoops to be sure that she would be 100 percent legal when she got here. That kept us apart for six months after the marriage, because I was in the U.S. working and she had to wait in South America until her green card application was processed and approved. To see hordes of illegals who think they have a "right" to skip the system, ignore the laws, and just walk in really [irritates me].