New York City
New York, New York. We hear it's a helluva town. (Actually, we hear that from the same guy who gave us West Side Story). Here, it's a nice modern stand-in for Shakespeare's Verona, right down to the heat rising from the streets and the nice young men trying to kill each other.
New York's the perennial modern metropolis, the biggest city in America, and one of the political, economic and social centers of the world. Yet even here, there are still corners where things aren't fair or right, and where people still fight over tiny little scraps of dignity.
The setting comes at least partially from the fact that everyone who created West Side Story was from New York. They wrote about what they knew, and what they knew were those working class neighborhoods full of kids with not a lot of hope. What better place to talk about rivals who hate each other's guts and a nice young pair of kids who think that somehow, some way, their love can get them out of there?
The setting was so important to the tone of the film that they actually went to the streets of New York to shoot it. A lot of the outdoor scenes took place on actual streets that you can still visit today (although many of the buildings were torn down shortly afterwards to build the massive Lincoln Center for the Performing Arts). Robert Wise was chosen to direct in part because he could shoot scenes like that very convincingly.
The effect was to make the story feel very real and grounded. That helps keep the musical numbers from flying off into the realm of fantasy and lets us know that the ideas they're talking about—and that William Shakespeare talked about before them—still have relevance today. Hence, New York: city of cities where the good, the bad and the ugly all meet up on those hot summer streets.