Basic social services are designed for the public good. If they cost something, that's because we're getting something back. As a quick illustration, those "overburdened emergency rooms" do serve to stop illness from spreading through the population, and do help get people back to work. They do not cause illness. Public health is for everybody's benefit. Welfare, for which aliens don't qualify until they've been here a while, is another, and has a specific purpose and (since Bill Clinton's time) definite ending dates) is another: but welfare for aliens is substantially cheaper than citizens (see the study.).
As a quick shorthand, let me put it to you this way: every mouth comes attached to two hands. For further development of the idea using rigorous logic, see Julian Simon: greater population leads to greater general prosperity. (Not a controversial idea in economics).