I think that we are tilting at windmills if we try to stop the flow of illegal immigrants by closing the border. They will tunnel under (which they are doing now), take boats to unprotected beaches (which they are doing now), travel to Canada and cross from there (which they are doing now), get smuggled in hiding in the bottom of a truck (which they are doing now) or any other of a hundred ways they can get into the country. There is no way we can protect the thousands of miles of border that we have.
My solution is to make it so they don't want to come here. I would pass a law that says that if you employ, rent or provide housing, sell a vehicle, provide healthcare or provide education to any undocumented person, you could be subject to a fine of $50,000 per undocumented person. And, I would provide a "bounty" that would pay $25,000 to anyone who turns in a person doing any of these things. If you are "legal", you will have the documents you need to exist and live. If not, go home. This also would provide a means for checking visas for those who came here "legally" then just stayed. If your student visa expires - no education, job, etc. If your temporary work visa expires, no job.
That way, even if they come into the country, they will not be able to find work, not be able to find a place to stay, not be able to educate their children, buy a car, or anything. So, why come?
JMO