Thankfully, it is perfectly legal for a citizen to walk down the street in America without carrying any papers, so not having any papers does not prove anything.
Prison population too high? Fix this "supply side", build much much bigger, much cheaper gulags than today's fancy, small prisons - U.S. has a warm climate, that won't be hard or expensive or too torturous - and make sure criminals and delinquents know that these will fit EVERYONE. Make them work (say build roads) to pay for that.
And yes, U.S. needs a 24/7 physically defended border. Russia has it being a much poorer much sparser populated country with a much longer border, so U.S. can too. No "refugess", "asylum seekers" etc. You try to cross border without papers or outside of designated crossing points - you get a bullet.