How can the play be considered a tragedy?
Aside from the unity of place A Streetcar Named Desire doesn't show too much fidelity to the Aristotelian theory of drama. But in the character of Blanche Dubois there can be no doubt that we are presented with a tragic heroine, brought low by a fatal flaw, and invoking pity and fear in the audience along the way. All the traditional elements of the tragic heroine are present in Blanche. First of all, she comes from a respectable family, one that enjoyed wealth and status. Yet, look at her now; and see how far she's fallen in the world. In Greek drama a tragic hero or heroine comes to grief through something they've done; they are not simply the plaything of fate. And in Blanche's case it's her self-absorption, her yearning need to be wanted and loved that leads to her sad demise.
Blanche is also sorely deluded. She thinks that purity of heart and inner refinement will be enough to wash away the taint of sin she's acquired from her numerous illicit liaisons. Yet try as she might, Blanche cannot escape the stifling restrictions placed upon her by a society that regards an occasional lapse in moral judgement as expressive of a fundamentally bad character. Blanche hopes that by starting over in a new town she'll be able at long last to find some measure of acceptance. But she can't; Stanley sees to that. By dredging up the details of her sordid past he makes sure that Blanche can never begin to experience the peace and emotional security she so desperately needs. A frail creature like Blanche is simply not cut out for this world, and therein lies the greatest tragedy.
A Streetcar Named Desire does not meet Aristotle's classical criteria for tragedy. But modern tragedies often delineate "what might have been," and this is evident in "Streetcar." Blanche DuBois was, once, a woman of refinement and learning. Partly through circumstances beyond her control (a world changed beyond comprehension from her sheltered upbringing) and partly through her own weaknesses, she has degenerated into the the manipulative, deceiving person we meet in the play. Her tragedy is that she could have been so much more.
It's important to note, though, that there are tragic elements present in all three other major characters. Stanley is not simply a brute, although he commits brutal acts. He is a person trying to protect his world, which is being threatened, and responding to taunts that hurt and provoke him. Stella, forced to choose between her husband and a beloved sister, makes a choice that will haunt her for the rest of her life. And Mitch, who rejects Blanche when the truth about her is revealed, throws away what might have been a chance for happiness for both of them.
The tragedy is the downfall and pitiable state of Blanche Dubois. Initially in the play, we see her as an interloper, interferring in Stanley and Stella's marriage and generally being a nuisance.
However, as the play progresses, she becomes more and more pathetic. We learn of her life as it faded from Southern Belle to near-prostitute, of her gay husband who killed himself after being discovered with another man. In desperation, Blanche tries to hold on to her illusions of youth and dignity, losing ground in every scene.
By the play's conclusion, Blanche has clearly lost all grip on reality. She is led away to an asylum, she makes a final grab at dignity, telling the doctor (who has just declined to put her in a straight-jacket) "I have always depended on the kindness of strangers."
It is not only a tragedy for Blanche. Stanley and Stella's world is forever shaken. More than ever before, Stanley realizes the divide in status between himself and his wife. Stella, for her part, has lost, probably forever, the sister she loves.
A Street Car Named Desire , does not approach tragedy in the dramatic sense. It contains none of the elements necessary to tragedy. Blanche arrives a sick woman and exits the same way. There is no element of discovery or self realization in any of the main characters, there is no tragic flaw which precipitates the fall of the protagonist and there is no reversal. That Stanley may now have a somewhat different relationship with his wife, hardly constitutes tragedy. Streetcar is a wonderful play, and Blanche a somewhat sympathetic character, but that alone does not a tragedy make.