I think the 2000 mile southern border could be adequately controlled by as few as a few thousand border agents using mostly technology.
Start with a serious fence/wall (or whatever it takes) in populated areas, and a set of wires 50 yards apart in less hospitable areas.
Less hospitable areas can be patrolled primarily by armed drone aircraft controlled from remote stations. A drone can easily see 2 or 3 miles so could patrol a 5 mile section of the border. At a couple million each, you could do it for a billion or two, including the ground stations. Can staff them with another few thousand pilots and ground support people. Can probably run the whole thing for 5-10 billion a year.
Someone violates the border, a warning shot is fired in front of them. If they refuse to return to their side of the border, something nastier happens. Maybe tear gas. But if that doesn't work you have to get nastier yet.
Once we have control of the border, and have deported the illegals, we can work out some kind of guest worker program if it is truly needed.