We can. It will require a 20 foot reinforced and electrified barrier with razor wires around the border and patrols by armed soldiers with night vision goggles. It will require the gunning down of families that escape to our country, if they refuse to stop or they don't understand the words "halt and put your hands in the air or you will be fired at." It will require a dictator who can ignore the pleas of his countrymen to "have some compassion" (you may want to forget America's reaction when Elian was sent back, but it doesn't change the political fact that most Americans have hearts and souls.) It will require the Patriot III-IX acts, incredible violations of our privacy, a huge government database, and a national ID.
Closing the borders is not worth the price. It would be better to do what Dubya has proposed. Like Reagan, Dubya has proposed amnesty for working families, tougher enforcement of current laws for criminals, increasing the flow of legal immigration, and free trade to prevent families from wanting to leave Mexico. It may not be a perfect solution but it is far better than closing the door to the shining city. I am sure that your predecessors would agree.