Let's say poor surrounding nations could pour into America whenever they please, is there enough work there to support them?
Ok so then is encouraging them to make a dangerous trek into a country where they will be left to starve to death anyway a kind lure?
Ultimately the problem is Capitalism
There's coming a time soon where Capitalism won't be practical to sustain the majority and they'll have to look to a new platform
Outside of the concept that everyone must work in order to be allowed to live there ARE enough resources in this world for everyone to live comfortably
It's just that some will work and other won't
In order to address the human heart that say 'what is fair' is more import than another human's life there are mediums that could still be implemented to balance give/take
One such medium is to lock Citizenship in America. From here on out the American citizens in the country and their offspring will be the last to get citizenship
Secondly the country shares enough resources with everyone whether citizen or not so that they can live, at least, a poor class lifestyle.
Thirdly those who do work can earn more on top of their pittance to spend on and enjoy the works of others.
Fourthly hiring non-citizens is completely illegal with a hefty penalty; taxes raised slightly more to implement more government auditors to ensure that no one is earning under the table - especially non-citizens.