Tax these transfers, and tax them heavily.
You can't tax them. Three states have proposed that (Georgia passed them). You then legitimize illegal immigration forever and governments have an incentive NOT to secure the border.
No! If it can be shown that the transfers are illegal, you confiscate them, you don't tax them.
We shouldn't be taxing money transfers because some portion of them are illegal. That ends up encouraging our government to not address the problems, because addressing the problems costs them revenues.
Taxing wire transferrs to attempt to address this is a very, very bad idea.
Require identification. Make it harder to send illegally earned money this way, but taxing it is a horrible idea.