It's not illegal to apply for asylum at the border.
It is to do it fraudulently. "Asylum" is just a ploy to facilitate illegal migration, just like in Europe. You and other open-border types know that as well as anyone, but won't admit it.
In America, people are considered innocent until proven guilty.
If you can prove beyond a reasonable doubt that an asylum seeker lied on their application, then you can say they did something illegal. But if all they did was show up at the border and apply for entry, that's not a crime no matter how much it upsets some folks.
I agree that we have to modernize our immigration and work permit system, especially considering how easy it is these days for people to travel thousands of miles and cross multiple borders. But I don't see immigrants as a threat to our way of life or destined to be burdens on our society. They're just people like my grandfather, coming to America for a chance to improve their lives and that of their children. And we have plenty of room, especially where I live.