Ludwig von Mises has a brilliant argument for open immigration policies in a section called "Freedom of Movement" in his book Liberalism. While open immigration in a liberal world is definitely something to fight for, we unfortunately don't live in that world yet.
Our country and the world would benefit from increased immigration, but not if we extend free health care and other benefits to immigrants. In our current welfare state, each immigrant becomes a burden instead of an advantage. Furthermore, those most likely to immigrate are not the hard workers who wish to take advantage of America's freedom and economic opportunities. Instead, it is those looking for a free ride on America's generosity.
In an ideal world, in Mises's liberal world, open immigration would make us all better off. But in our current welfare state, allowing open immigration would bankrupt our country.