You put troops on the border, you will eventually have controversial deaths, whining liberal groups, hand wringers, and chaos of one kind or another. Hard feelings on both sides of the border will escalate. The perpetual cost of maintaining troops on the border is huge. Not a win win situation.
A fence alters the current situation. It is no longer easy to cross - it takes an overt physical and criminal act. Nobody can claim he "just blundered" across the border. It makes those that attempt to cross think "these guys up North are really working to make it dificult" - as opposed to the present - "hey Paco - its just a short walk and your in." A fence changes their mindset!
Now you team the fence up with strong border enforcement for climbers and tunnelers (not necessarily troops but Border Patrol)....with aggressive "rules of engagement" and pretty soon the folks south of the fence get the idea that "it ain't happening no more."
Then, and only then, do you have border control.
Yes!! You understand that any real solution will have to have more than one step. But step one has to be a serious fence/barrier.