The American public isn't willing to spend the money. Period. They'd rather have it spent on pork in their districts.
Second: absent immigration (legal AND illegal combined), America's population will decline. The only way to avoid this is to draft women into the Maternity Corps and order them to have babies, and after they reach majority, draft their kids into the Border Patrol. In short, unless you're willing to do those things, your policy is not sustainable in the long run (i.e., more than 20 years or so).
I have seen no polls on this, so I don't know. But if you'll allow me to extrapolate from the polls showing 65-85% of Americans being against unfettered illegal immigration, I suspect there would indeed be a great deal of support for beefing up the manpower on our borders. Unfortunately, no politician in either party, except Tancredo, is even proposing this.
Second: absent immigration (legal AND illegal combined), America's population will decline.
I'm not sold on the idea that only the illegals can save us from a population decline. There are millions of people around the world who have applied for U.S. citizenship. We should bring these people in faster.