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.
Then I'm sure you'd open your doors and welcome and feed and clothe a few dozen of these asylum seekers into your home!
Been there, done that. Six times. Not for asylum seekers specifically, but for people coming to Alaska to work who needed a place to stay rent-free for a few months while they got situated. Also loaned a car to some Bulgarians here on H-1B visas last summer, and am currently providing rides to some Jamaican H-1B visa workers when the weather's bad or they can't get to where they need to go on their bikes. Also provided a nice down jacket to one Jamaican girl this year (she was freezing in our spring weather), and have bought coats and food for others over the years. My husband has a knack for finding cars, tools, and whatnot on Craigslist for very reasonable prices, to give free of charge or re-sell at no profit.
It's not hard to be helpful and in my experience, most people only need an occasional boost so they can succeed.