I suggest machine gun nests along the border, and a Constitutional amendment allowing surviving invaders to be sold as slaves.
Or, if our low information citizens are feeling terribly concerned about the people we use as a convenient underclass, I'd be satisfied if we merely annexed Mexico.
But I invite anyone who lies in a fantasy world of open borders and unicorns to allow illegal invaders to live in their home rent free while supporting them.
Quod scripsi, scripsi
The opening question should have been: Should we close the borders? How do you open something that's already open?