Citizen groups have tried to do exactly that, and have been called racist homicidal maniacs by both governments. And that's without actually shooting anyone.
American landowners along the border who are robbed in their houses almost weekly who have shot and wounded intruders have been charged with assault etc. and are facing huge civil suits put on by "civil rights lawyers" from both nations.
Both governments have WANTED the de-facto open border, at least until 9-11.
1) A virulent ideology of self-hatred that has taken over our educational system and our mass media/entertainment.
2) Uncontrolled immigration from non-Western parts of the globe which have an unfortunate combination of exploding populations and crumbling economies.
Simply put, massive population growth is occuring in failed third world societies...and these people will do anything to escape and come to Western societies where Westerners have created stable republics with free, growing economies.
The West rose once before in the Classical period, only to be destroyed by a combination of social decay, bloated government, and mass migration by barbarians. It appears that the second rise of the West will meet the same fate.