What makes the US so unique is its social heterogeneity. Indeed, opening up to the rest of he world is the easiest way of assimilating the best the world has to offer. The US has been the most liberal in embracing the global culture, paving the way for other civilizations to follow suit.
But every political union has its downside. A non-porous border is one of them. While in a utopian world borders shouldn't exist, in the real world they should...at least as long as the global socio-political scene doesn't change dramatically.
illegal immigration is an issue that needs to be tackled with utmost care. It really doesn't make sense to embrace all and sundry with open arms, nor should the US spurn all advances.
It all boils down to policies that best suit the welfare of US citizens first and foremost. I'm sure there are enough policies that deal with this issue. If they need be fine-tuned, so be it; but let the illegal immigrants be deemed so...and sent back home. If America needs them, they can always be invited back...as respectful foreign workers.