I don't agree with stopping people who look like illegal immigrants (aka brown skin) to check their papers. Sounds very Orwellian.

U.S. should instead build a high, steel fence across the entire southern border, something that isn't very thick, just sturdy and tall, to keep down costs.

Put sensors on the wall, and deep into the earth, so if the wall is breached or a hole is being dug, you can dispatch officers to check it out.

Once the wall is built and monitored, you can start a naturalization process for illegals already in America, given a 5 year probation period.

Even Mexico will generally benefit from a wall:
1) Drug cartel problems in Mexico will reduce as the drug trade will be too expensive to smuggle into the US
2) Gun smuggling back into Mexico will dry up as well.
3) Mexican tourist border towns will revive as the Drug Cartels dry up.

Though Mexico will lose out in:
1) Not as much money being sent back to Mexico
2) No place to send their poor and discontent which might lead to political instability.