If by "purpose" you mean what argument A Streetcar Named Desire is making, I think there are several possible answers. One of the main themes of the play is the difference between fantasy and reality: Blanche lives in a fantasy world that is at odds with the "reality" of Stanley's life in New Orleans. Blanche is nostalgic for the South of her youth (although that also might be a fantasy). Her sensitivity and cultured nature are things she has preserved about herself from this treasured past. Stanley, on the other hand, represents a working class point of view that simply can't understand Blanche.
I would argue that these two characters are not that dissimilar. Both Blanche and Stanley are the result of an economic system that has destroyed the old aristocratic South and has turned Stanley into a brutish wage slave. If anything, the purpose of the play is to show the human costs of these changing socioeconomic forces during the post-war era. While Blanche has the manners of a Southern lady, she is actually destitute and has to rely on the charity of other poor people. This inversion of economic power lies at the heart of the play's conflict.