The film's director, Pete Docter stated in the accompanying Artbook that Monstropolis is a parallel universe to our own and that monsters have been around for a considerable amount of time. No mention is made to their having lived in our world prior to the building of the monster cities but given that door technology appears to be relatively recent, presumably there were earlier (less reliable? less sophisticated?) ways of traveling between the worlds:
"In 1997 we began to design the parallel universe we called Monstropolis. While our development team was working at Pixar, we also commissioned some of our favorite children's book illustrators to do inspirational designs. After I gave them a brief summary of the story, each illustrator was given freedom to design his or her unique vision of the monster world."
Monsters may creep out of our closets, but they don't live there. What do their homes, their cars, and their cities look like? Early on. the sky was the limit. But as we thought about it more, a few facts came to influence our designs:
Monsters have been around for as long as there have been frightened humans. The city's design should reflect this long history.
Quotes from The Art of Monster's Inc: Monstropolis