I would argue that the problem is that it's already too difficult for people to cross the border legally. If immigrating and gaining citizenship was more viable, fewer people would have to take these sort of risks to try and get into the country.
We don't have Canadians freezing to death in trailers trying to get into the country illegally, and it's a hell of a lot easier for Canadians to cross the border and enter the States.
If the wall were built, this sort of human smuggling would be more common because it would be impossible to make the trip by foot.