I think that Williams' narrative can be seen as a tragedy for a couple of reasons. The first is that it sets Blanche against conditions that are larger than herself. Blanche acquires a tragic figure sensibility when she is set against the changing times from Belle Reve to the modern South. Blanche is almost helpless in how life and social values have shifted, leaving her behind. In this, she acquires a tragic sensibility.
If we operate with the belief that a tragedy in the modern sense is a story "where some of the characters’ lives or story arcs end badly," then Blanche is in a tragedy. The arc of her life is a tragic one. She enters New Orleans with the hopes of finding someone and leaves as dependent "on the kindness of strangers." She is abandoned and raped, discarded and abused. While Blanche bears responsibility for what happens to her, I think that Williams' depiction is one where it is difficult to say that she deserved what happened to her. His construction of her is a sad one, reflecting in some of the worst ways what it means to be human:
... when I think about her, Blanche seems like the youth of our hearts which has to be put away for worldly considerations: poetry, music, the early soft feelings that we can't afford to live with under a naked light bulb which is now.
In seeking to make Blanche such a representation of what it means to be human, I think that Williams makes her character tragic. It is here in which the drama can be seen as a tragedy in the modern sense.