Usually when a city has two professional sports teams that play the same sport, there is some sort of geographical divide that determines whether a person is a fan of one team or the other. In New York City, generally if you're from Queens, you like the Mets, and if you're from the Bronx, you like the Yankees. Mostly because Yankee Stadium is in the Bronx, and Citi Field is in Queens. But Fenway Park and Braves Field were located barely a mile away, in the same neighborhood of Boston. So what kind of person supported the Braves?
The Braves had more of a working class reputation. The Red Sox tended to be the better team with the better players and their ticket prices reflected that. The Braves were the cheaper option. Now obviously, please understand that this is a generalization and not a hard and fast rule. As an interesting historical aside, here is a relatively interesting article about Boston's remaining Braves fans--but do know, it does not address your question. Fun read none the less.
http://www.sbnation.com/2011/4/23/2128257/bostons-lost-cause
I think your premise might be a bit flawed from a historical point of view. I'm not debating whether your premise in New York today is correct or not so much as the reflection of the way New York baseball was before the Giants and Dodgers moved to California.
You had much the same set up described with the Polo grounds and Yankee stadium. I don't have any sources that explore this, possibly Ken Burns does to a certain degree. Anecdotally there was a sense that the Yankees represented a more elitist ethos than the Giants did but I'd love to hear a historian weigh in.
Note that there is a similar set up between the Jets and the Giants in football where the cross town animosity is possibly not as significant as it is demonstrated in baseball.