It's abundantly clear from the opening scenes of The Glass Menagerie and A Streetcar Named Desire that the families depicted in the respective plays are quite low down on the social ladder. They may not be dirt poor, but they're certainly not especially prosperous, either. If one had to put a class label on them, one could say that the Wingfields and the Kowalskis are lower-middle class.
The Wingfields live in a tenement building that faces onto a shabby-looking alleyway. This is clearly not a part of the world where anyone would want to live if they had any choice in the matter. As if the set design wasn't enough to tell us that times are tight for the Wingfields, Tom's monologue, with its reference to the effects of the Great Depression, makes explicit what kind of economic environment his family is living in. This is a family very much down on its luck.
Much the same could be said about the Kowalskis. They live in a fairly run-down part of New Orleans—not quite a slum, perhaps, but still pretty downmarket all the same. The apartment block in which Stella and Stanley live has clearly seen better days, what with its shabby exterior and faded stairs.
In the opening scene, we hear the sound of piano music drifting in the evening air from some dive-bar in the vicinity. This only adds to the sense that this is a somewhat disreputable neighborhood, certainly not the kind of place where we'd expect to see the snobbish and refined Blanche DuBois.