I would challenge this. If you do believe that there are violent criminals coming through the porous border, whether it's 1% of the illegal immigrants or 100% of them, trying to seal the border off is not irrational. I'm not endorsing the physical wall itself, as I know a ton of illegal migrants are just overstaying visas, and I've heard of ladders and tunnels.
I think what's really compelling, and what the Left can't seem to relate to, is this: Everyone serious does believe the true fact that illegal immigrants have a lower rate of committing crimes than the overall population. But people who are victimized by those crimes have a valid point that those crimes are still incremental crimes - meaning that if we already had 1000 people in $BORDER_STATE who are going to commit violent crimes, letting in 1000 more people, even if only 10 of them (1%) are violent criminals, gives us 1,010 violent criminals. That's more crime than we had before. It's not like we get to trade in 10 of our own criminals for 10 immigrant ones.
Making no effort to control who comes here is irresponsible, because of course if there's a country that doesn't even try to vet you, and would feel guilty making you leave, of course criminals would be excited to go there.