Sure secure the border. But how are you going to deal with the scofflaws that are already in the country? Are you just going to start rounding up folks with brown skin like me? Of course not. We already require folks provide proof of citizenship for a host of things. Why not make an ID that cannot be forged and require that it be presented when one gets employed in the US?
Once the inflow is essentially cut off, the problem of those remaining in the country won't seem as severe. They can be dealt with at a more deliberate pace.
Heck, given the often ugly ferocity of the debate around these parts, some of the hunt-'em-down crowd might be penciling you in on their lists right now. ;)