Truth is, a lot of the programs that might help stem the flow of poor migrants, like the Peace Corps and/or USAID programs, are ones that the current regime has cut to the bone. Seems as though the best way to help with the issue is to work with the countries to help combat poverty (for example).
The cynical side of me is that many of the extreme Republicans secretly want the borders open, not only for cheap labor but so they can continue to use it to rile up their voters.