There are some direct relationships between birthrates and immigration rates. But it is also true that in economically prosperous countries where there are high levels of prosperity and education (which the US still is, despite recent downturns), immigration can be a substitute for lower birthrates to ensure continued growth. Setting aside sustainability issues (i.e. is growth even the right way to go, in the long, long run), and ethical issues (i.e. my long-declared position that immigration is, in fact, a human right) immigration still becomes a critical factor in determining an advanced economy's health.
Apparently the US birthrate has recently plunged. No one is sure what exactly is going on – it's tied to lower immigration rates (which in turn are tied to the poor economy and high unemployment), but there seem to be other things going on too. Ezra Klein at the Washington Post writes:
A key contradiction in American public opinion is that many people simultaneously think that immigration is bad for the economy (“they’re taking our jobs!”) and that a low birthrate is bad for the economy. But they basically lead to the same economic problem: too many old people, not enough young people.
This really does capture the cognitive dissonance behind anti-immigration thinking.