There is something else we would need: an inexhaustible supply of 18-year-olds to draft. Just putting ten people per mile on the border on a 24/7 basis would require over 100,000 troops--not counting cooks, bakers, clerks, logisticians, armorers, and command staff--and 10 soldiers per mile isn't nearly enough. 100 per mile is a bit closer to reality. That's "only" a million troops. And, given standard US Army manning for combat troops to support personnel ("tooth to tail"), that works out to about 10 million troops in the total force package. If "Army Transformation" works as advertised--i.e., if a bunch of technologies that haven't been invented yet arrive on schedule--that figure can drop down to about 6 million--but that's in the year 2022, and you're trying to solve a problem NOW. Of course, you can always move those personnel "under the line" or "off budget" through creative (read: dishonest) accounting techniques. But you still have an irreducible minimum force package of over 1,000,000 infantrymen plus their support.
We don't have 1,000,000 infantrymen available. You're going to start yanking people out of other jobs, and that will convince a lot of folks to leave the military, whereupon you'll need to have a draft.
Once you've closed the southern border, you have to secure the coastline and the northern border--because when you nail the front door shut and ignore the windows and back door, you'd be amazed how fast the scumbags figure out that that's the way to go. So your force package probably exceeds the total manpower pool available in this country unless you start drafting illegal immigrants.
Let's also not forget that when you have people conducting anti-smuggling ops, a certain number of them will go bad. So far, the military has an admirable record vis-a-vis the police force. However, please do not confuse a lack of temptation with innate moral superiority.
However, the price tags are very different. The price tag for a corrupt police force is an increase in crime. The price tag for a corrupt army is the destruction of America.
A better long-term solution is to impose "regime change" on Mexico.