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.
They need to seriously reform immigration ---actually many non-working types come in by using family sponsors who only need to be 125% over federal poverty levels ---of course they only need to find ONE such relative, the rest can be quite wealthy and then the sponsor's income is considered in with theirs when qualifying for SSI, Medicaid and many other handout programs. Or many come over and have anchor babies which then qualifies them to stay and collect welfare, use our hospitals, schools and the rest. So many immigrants use this system and collect welfare while and at the same time it can be quite difficult to someone who is willing to leave their family back in Mexico to just come and work, but the only thing they intend to do is work for any dime they get.