The border is not open. We’ll get nowhere if people continue to fall back on hyperbolic rhetoric like that.
The immigration system is broken, but the best chance we had of fixing it, back in 2013, was scuttled not by Democrats but by Republicans like Ted Cruz.
A combination of cracking down on domestic employers and generous aid to developing nations would go a long way to changing the incentives that bring people here, but good luck getting conservatives to agree to that.
Not sure the Denmark model would work here since we’re a federal republic, and states (unlike the federal government) generally can’t discriminate on the basis of citizenship or nationality without running afoul of the Supremacy Clause.