The Border Patrol and Customs (in the Department of Homeland Security) is actually responsible for securing borders. States do not get to decide how their borders are enforced.
We all have to deal with illegal immigration, most of which doesn't even happen on the southern border. We manage to do it without building death traps or tricking people into getting on busses and sending them to states we don't like.
I know Texas and Abbot think they're special but they're wrong.