What we know about the atmosphere of setting in Streetcar comes from Williams’s stage directions. He tells us that us that the area is "poor" but "has a raffish charm." He says the sky is "a peculiarly tender blue, almost a turquoise, which invests the scene with a kind of lyricism and gracefully attenuates the atmosphere of decay. […] In this part of New Orleans, you are practically always just around the corner […] from a tinny piano being played with the infatuated fluency of brown fingers. […] New Orleans is a cosmopolitan city where there is a relatively warm and easy intermingling of races" (Stage Directions, Scene One). Here’s a great image of this mood-setting coloring, by the way.
This introduction – and particularly its attention to social context – is important for the way we read Streetcar. Race relations weren’t "easy" everywhere in the 1940s, but it’s important to establish the atmosphere in this particular setting, especially since Blanche brings to the Kowalski apartment her prejudices, which prove to be out of time and place. Class distinctions don’t matter here, which is why Stella and Stanley seem to make a fine match despite their backgrounds.
As far as the actual physical set-up on the stage, it’s important that we can see the upstairs, the downstairs, the interior, and the exterior. The play’s action takes spectacular advantage of the flexibility this offers, whether it be Stanley listening in on his wife and her sister, Stella walking down the stairs to the waiting arms of her husband, or the way we get to watch two scenes at once – Blanche flirting with Mitch in the back-room while the men play poker in the front.