A popular rebuttal is that immigrants will contribute to the growing economy and will pay taxes. But we are talking about completely open borders here (anything else is just some fiddling around current system), where anyone willing to do even minimum wage job can migrate. How can such people contribute enough in taxes, when the welfare cost needed to support them (schools, roads, sewers, hospitals...) and their families far exceeds the taxes paid by them? And while I admit that I don't have any stats to support my argument, I am just going by my gut feeling that generous welfare is usually paid for by the richest and not by the poorest.
Another offered solution is to cap welfare for new migrants until they pay enough taxes. But that just creates a two-tier society and is a separate can of worms in itself.