How can the play be considered a tragedy?
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.