You may want to consider the difference between "is for" and "built for." We can have philosophical disagreements about what the military "is for." However, nobody who is familiar with the development of the post-WWII U.S. military can claim that any part of it (besides the Coast Guard) is "built for" domestic border defense.
I actually agree that some branch of the military SHOULD handle this, but no existing formations are appropriate. A militarized Border Patrol with members subject to UCMJ and trained and organized specifically for patrol duties would be the best solution. And again, these people realistically need to have law enforcement duties which makes Department of Homeland Security control more (legally) appropriate than Department of Defense control.
Border patrolmen should be a great deal easier to train. Although you have to teach them all Spanish, you DON'T have to pay for combined arms life-fire exercises, deployments to training centers, etc.