First, as much as one might sit and think the Border Patrol has the strength to monitor the border.....they don't. They'd probably have to add at least 50,000 additional personnel to their manpower to reach some reasonable stage of border control.
Second, there's some people talking of the idea that a 'peace-dividend' might come out of this South Korea-North Korea talks, and that the 37,000 troops in South Korea might not be necessary. I think that's a minimum of three years away, but if some treaty occurred....then you'd have 37,000 troops coming back to the US. What exactly do you want to do with them?
Third, if you were of the mind to cross the border, and got to the Trump-fence or Trump-wall.....maybe you might still have enthusiasm to cross. If you figure in the odds of three-hundred Army guys along a sixty-mile area that you were attempting to cross....the odds are against you.
Fourth, there's all these recon assets that the Army would bring with them. It would reshuffle the whole border crossing strategy.
There are negatives to this idea....for example, we aren't talking about 3,000 Army personnel. We also aren't talking about the Border Patrol going away. I also don't think the Border Patrol would like competition.