This is venturing a little off topic from refugees to illegal immigration now, but certainly the lax immigration stance of the U.S. government in the last generation or so has been one of the big factors in driving down U.S. wages and decreasing the average standard of living.
You realize that the net undocumented movement into and out of the US has been effectively zero (technically, a slight net loss) for about the past decade, right? It’s basically like the entire Obama administration was an experiment in what happens when there’s zero new illegal immigration. It’s very much like a big wall was constructed, except much cheaper and actually effective. Zero can be a big factor, though, for extremely small values of big.
The big increase in illegal immigration that was triggered a few decades back (the tail end of which lasted into this century–so your “last generation” comment could be referring to part of this) was because the Reagan administration severely tightened controls. What used to be relatively lax movement of seasonal workers back and forth got dramatically restricted, and those seasonal workers eventually had to choose which side they’d rather be stuck on–and they chose the US side. So tightening immigration policies is actually what led to the last real boom in illegal immigration in the US. Not that we don’t have people hard at work fabricating a more recent imaginary boom in illegal immigration due to an imaginary lax immigration stance.
Economists generally agree that the effects of immigration on the U.S. economy are broadly positive. This includes refugees, so perhaps the household metaphor is too simplistic.