Debunked?
Then Sheriff Larry Dever of Cochise County, AZ is lying - http://cnsnews.com/news/article/71105 (with video of him saying this.)
Dever, a native of Cochise County, has been in local law enforcement in the county for three decades. He was elected the county sheriff in 1996.
He is not Border Patrol.
He is not lying (deliberately misrepresenting what he knows to be true), but he is saying something that is not true. He may believe what he says but that doesn't make it truth.
He is right that it is dangerous, no kidding, but there are NO do—not-patrol areas for the Border Patrol and the Border Patrol has not been directed not to patrol areas.
Check the link I provided.