An economy with a welfare state can't afford to import low productive worker...
They could make laws that would make it harder for new immigrants to obtain welfare. Such as giving out a lower form of citizenship for the first 5 years. A green card of sorts.
You don't know how politics works in the US. You have to promise more to them, not promise less.
Over-promising, that's what got Obama elected. That man had lovely posters and music and made it seem like unicorns and pots of gold at the end of the rainbow were real.
Republicans are losing because they're not over-promising (for the most part). They may be bat looney, but they're not doing what it takes to win.