Revamped work visas, Mexican stability, and ameliorated crisis in Central America would take care of it, but even then you have to recall that illegal immigration by land is at an historic low, so it's mostly an issue of Central American refugees and work visa abuse (often an employer-side abuse rather than a choice by visa-holders; shocking how a work visa scenario can turn into a human trafficking scenario). And the Central American refugees - they are indeed refugees - are entirely our problem. We made this.
So how could a desire to create an environment of fear and impunity for mostly-Mexican aliens who have been residing in the country for decades - how could it be fueled anything other than hatred and spite? Someone who declaims illegal immigration should designate long-term resident aliens as the least priority. But a lot of Republicans really just want to see these people suffer; it's not even about policy to them.
Punish abusive employers, restrain the big corporations, and repatriate individuals, all prior to any comprehensive reform. As for the rest of the aliens, if you can't accept outright amnesty at least take a moratorium on harassing them.
Edit: Also important to keep in mind, the 1996 Clinton immigration reform, partly by making crossings more difficult and dangerous, is what established the massive modern population of long-term unauthorized residents. And that's while stripping the target population of various rights and procedural remedies, and making the process more abusive overall.
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