I guess it is in a way because we tend to think people who came always stayed but then again it is hard to go to another country and culture, learn a new language. Without the safety nets we have now though many Europeans and others went back ---which back then meant another long boat ride ---not easy. Without welfare, many of today's immigrants would also go back, we see many now who won't learn the language of this country but now they don't have to because they can just depend on taxpayers to provide for them if they don't care to make the effort.
The political will to cut off welfare to immigrants today would never be there, all we'd see on tv are stories of hungry children and the homeless. That's why the libertarian's position that we can have open borders if only we did away with welfare is unrealistic.
It's not a matter of ending immigration, but rather focusing on those who have skills and are employable. Even then, it shouldn't be so overwhelming it displaces American workers as H1-B visas are doing.