Well, you’d originally written “deal with them humanely, or to simply prevent them from crossing by any means, up to and including lethal?” And I figured that framing left more possibilities open than, well, This-Or-Not-This.
The 13th Amendment says that neither slavery nor involuntary servitude shall exist within the United States except as a punishment for crime whereof the party shall have been duly convicted. I propose that we make that a punishment for illegally crossing the border; we don’t have to go straight to it, we‘d still have the option of letting them off with a warning as we kick them out of the country — but once we’ve made clear that they can be sentenced to hard labor, we can amiably shrug if they mull it over and decide, oh, yes, that’s what I’d like, please and thank you; here I go walking right up to the border and, upon being given fair warning, accepting the offer.