Compromise
There are so many outright rediculous points in this thread, that it would take too long to address them all.
We need look no further than your post:
"How can you deny them the right to a better life?" Sophomoric statement of them all. Using this logic, it is now the responsibility of the US to seek out all individuals throughout the earth that have a lesser lifestyle, (say for example, the Iraqis, who are stuck in the bronze age) and bring them here so that we can provide for them a better life through free healthcare, welfare and education.
Your absurd manichean view is just nonsense. Under your logic, we'd better end all charitable foreign aid, all UN programs, all assistance and the Peace Corps, because any of those activities obligates us to do the complete elimination of all problems in the world as you said. Ridiculous.
The truth is that it's our erring in the other direction that is our problem most of the time. Our 'wanting it all on our terms' happens too much.
I already mentioned an example of history directly in Mexico for our own interests at their expense, but let's look at some others. Go back to the 1840's if you like and ask which of our principles respecting the right to self-determination we were expressing when we sent gunboats to Tokyo to tell them 'open up to our markets or die'. Which at the end of the century when we overthrew the government in Hawaaii to dominate it.
When the Iranians wanted to get a fair price for their own oil, the natural result of a democratic leader coming to power and representing the people, our response was not to pay a fair price, but to end democracy in Iran and put in a dictator, creating a brutal police force for him for killing and torturing. I've heard there's been some 'blowback' from that (the term was actually invented by the CIA for that incident).
When Chile elected a leader who wasn't a pawn of our corporations, Nixon ordered him gone. Elected Allende dead, dictator Pinochet in.
Similar for Nicaragua when they overthrew the dictator Somoza family who were 'our boys'; we trained, funded and sent terrorists to kill people, destroy their infrastructure, and terrorize the population into our ultimatum to vote out Ortega, even while we said Ortega would not respect the election if he did not win (he left office peacefully).
When the Venezuelans had the same thing as Iran, we tried the same result, only this time the Venezuelans reversed the coup.
When Ho Chi Minh came to use and said all he wanted was to be free of being a colony and asked our help because he liked our constitution so much - he wanted rid of the French, then the Japanese in WWII, then the French yet again after WWII - we put our ties to the French ahead of principle and told him no. That cost us a bit, too.
A century ago, when our ally Columbia, who we had supported against some rebels wanting to form an independant country Panama, told us we could not build the canal we wanted - we sent the troops in and told them that now, there would be a Panama, who of course was happy to give us the land.
You get the idea. We've had some problems where we are not exactly 'fair' and 'compromising' and living up to our principles in respecting other nations.
We've also done huge good at times. Which do we want to do now? Well, we can try to do *something* to help the situation in Mexico so that they become a more prosperous nation and have less need to get our jobs and drag down our wages, which is even in the long term interest of our corporations, even if in the short term cheap labor is nice - or we can be short sighted as your post suggests and leave the problems.
If the latter, we'll see other problems, as we already do from 10% of Mexico's population being in our borders, to their high susceptibility to drug trafficers.
Why don't we instead do what's both morally nice, and very practical and in pretty much everyone's long term interest?