If you had to flee America for a safe refuge, you would have to prove your worth to any place you go--even sparsely populated Australia. Belize, for example, imposes a $25,000 fee and other restraints for immigrants.
One idea that would be reasonable is not allowing welfare recipients to bear children; if you do, you permanently lose all future welfare. However, a better idea is to stop paying people to produce children they can't or won't care for by eliminating welfare (the government-run kind) altogether. Ditto for any benefits for illegal aliens--no proof of citizenship--no benefits. And finally, repeal the 14th amendment; no other country is this lenient in granting automatic citizenship to children of illegals.