"Prior to the 1840's people didn't regularly cross borders"
Yes they did. Much of Europe was composed entirely of borders. Remember Italy was a mess of states until the 1870's, same as Germany.
"Passports were invented as a necessity to international travel opening up to the middle and lower classes."
But that's wrong. How could requiring a document (that the individual had to PAY FOR) to enter a country POSSIBLY be cheaper than crossing without it? That doesn't even make sense. Keep in mind that borders were completely porous, and only soldiers were blocked from crossing prior to this time.
"Because you cant imagine the chaos of hundreds of thousands of undocumented border crossings."
You're right, that's why the EU and US have descended into chaos. It's total anarchy, I tellz ya!
"I guarantee every solution you have to that is either going to be stupid, useless or a passport like system (or possibly all three at once)."
Well, considering that
felons can get passports from the US, and God knows where else, a more modern solution would be to simply run criminal background checks on people applying for visas (ie those staying for an extended period). Much better to have a private system (which would necessarily transcend borders and *GASP* treat everyone fairly rather than giving Westerners a golden ticket while completely fucking over the huddled masses yearning to be free). No more of these "stateless" people getting stuck in airport terminals or refugee camps. Treat people like fucking people. What a concept.