open borders improve the lives of everyone.
If you're worried about people on welfare, ban children, not immigrants. Four times more people are born in the US than immigrate, and babies can't even work or pay taxes for two decades.
Paying immigrants "off the books" isn't necessary if those immigrants don't have to fear being reported to the police by their employers.
People won't immigrate if the pay is unsustainably low. That's what they want to get away from.
Any concerns about immigrants and welfare can be solved by just making immigrants ineligible for wellfare for a few years.
also, check out this study: http://blogs.lse.ac.uk/europpblog/2017/08/16/evidence-from-denmark-how-eu-immigration-can-benefit-the-welfare-state/