I was under the impression that before people used to just get on a ship and sail to America. Land on Ellis Island, get processed and move on to build a new life in US. Isn't that true?
Largely. Two things killed that off -- the jet age (meaning that travelling to another continent is now easy) and the greater connection of the third world to the first world.
Open borders for a rich country just aren't practical any more, if they want to remain a rich country. Like osmosis, open borders mean that people will flow from countries crappier than the US to the US until the US ceases to be less crappy than any other country.