The Army is not designed to stop smugglers, nor is it designed to deploy for long periods of time without a clear purpose.
I agree with the long periods of time part. I believe, however that the Army would do quite well stopping smugglers if they ran exercises on the border. Between a proving ground over here for Apache attack runs, Kiowas running reconnaissance missions, and Tanks, Bradleys, and Strikers engaging in maneuver drills, it would be very easy for them to catch someone on Infrared in a prohibited area, and direct a few MP's in a HMMWV to find out what they were doing there.
I'm not suggesting the Army be there strictly for interdiction, but that they do the same activities as they do already at the National Training Center at Fort Irwin. They won't cover the entire border obviously, but they could drastically shrink the area ICE and Border Patrol would have to worry about.