"In fact, the current configuration of our land borders didn't come into existence until February 14, 1912, when Arizona became the last of the 48 contiguous states to join the union. Oklahoma had joined in 1907, and New Mexico January 6, 1912."
The borders of those then-US territories were no different than the borders of the US states now. As far as people flowing freely, criminals were pursued. I'm sure you've heard of Pancho Villa and Pershing's "Punitive Expedition."
http://en.wikipedia.org/wiki/Pancho_Villa_Expedition
Nice of you to completely ignore the main point while trying to steer the discussion into a useless sidebar. The stone cold fact is that our borders have been wide open throughout our history. To point out this fact does not equate with wanting out-of-control illegal immigration to contiue. However, we are a nation with no historical precedent for having closed, heavily fenced borders, so agitating for such is like spitting into the teeth of a hurricane. Can be done, but not without a lot of unpleasant blowback.
Oh, BTW...the borders of the United States most certainly did not extend to their present configuration before the various territories became states. The citizens of the territories could just as easily have voted to become independent nations as to join the union.