Tennessee Williams' play, A Streetcar Named Desire, is a drama. The term drama is Greek in origin, meaning "action." Outside of that, dramas are placed into subcategories. This play is what is known as a melodrama.
A melodrama is a play whose actions and plot are meant to appeal to the emotional side of the viewer. The characters tend to be over-dramatic and overly emotional which adds to the overall sense of the "woes me" mood of the play.
While the production of melodramatic plays was coming to an end when A Streetcar Named Desire was produced, Williams (assumedly) did not care. The fact that Blanche was a character recognized for her dramatic sighs, overly emotional outbursts, and inappropriate flirtatious nature can support the melodramatic characteristics of the play.
While I support the play as being one of the melodramatic nature, some critics have identified it as one which belongs to the genre of dramatic naturalism. Naturalism was a movement, initiated by French writer Emile Zola (Germinal). Naturalistic writers were ones who wrote about the power of nature over mankind. Regardless of what mankind would do, nature would always win. Naturalistic authors/playwrights tended to take the stand of a scientist watching an experiment unfold before them. They, the authors, would not "interfere" with the action; they would only "notate" what happened.
Therefore, one could easily justify that A Streetcar Named Desire fell into this genre as well. Given Blanche's desire to conquer all, change Stella's life, and break Stanley, the fact that she goes insane shows nature's power over her.
I am sure you could make it fit into any genre that you wish. You would simply need to justify your interpretation of why it fits.
It could easily fit into the genre of a Domestic Drama given the play depicts everyday life for the characters. For example, the violence which happens seems so "everyday" that the characters tend to ignore it and see no problem with it.
Given that Naturalism is an offshoot of Realism, you could easily make it fit there as well. Realism simply lessens the idea that nature is all powerful (as seen in Naturalism). Outside of that, they are very similar.
What about "domestic drama" or "realism"? DO you think it could fit into these categories?