Citizenship defined by where one is born, by territory, is not without its imperfections, but it best upholds not only our belief in equality but the need for a cohesive community. In ancient Greece and Rome, only children of citizens received citizenship because that was the most efficient way to maintain social distinctions in a society in which slavery and other forms of status subordination were accepted. (The U.S. confers citizenship on the children of citizens too in some situations, but territory remains important: In some instances, at least one parent has to have lived in the U.S. within a prescribed number of years.)
By contrast, birthright citizenship was established early on under English common law, a legacy of the medieval system of feudalism and reciprocal obligation. A child was deemed worthy of protection of the sovereign in whose territory he was born. In exchange, the child owed the sovereign loyalty. That reminds us that citizenship is not just about rights. It's also about responsibilities.
In the long term, it's in this country's best interest to absorb the children of those who have made their way here, and thereby to establish the reciprocal obligations of citizenship. We all know the adage that owners take better care of their residences than renters. The same applies to full citizens and their nation. The more residents of a national community who feel obligated reciprocally, the stronger the community.
The New York Times "Room for Debate" blog has several entries on the politics of the issue, including one by yours truly.