The El Paso Times had a story on the possible closing of the border back at the end of April (link). Excerpt follows...
Closing the Texas-Mexico border to prevent the spread of swine flu is an option, but one that is not necessary right now, Gov. Rick Perry said Wednesday.
What does that mean, "closing the border?"
In reality, "closing the border" would include shutting down sea ports, air ports, all rail and road crossings. It's not possible, in other words.
Border security is one thing, but "shutting down the border" 100% is just not reasonable. The swine flu highlights just how impossible that task would be.