This phrase is a gentile way of suggesting that the "stranger" she is addressing will take care of her, and the unsaid, ambiguous implication is that she will be "grateful" for such care.
The phrase is significant for several reasons. For instance, Blanche says this in the middle of a psychotic episode to a mental health worker who is leading her away, so the "whoever you are" part is telling, because she has dissociated from reality. The part about "relying on the kindness of strangers" is ironic as well. Have strangers been kind to Blanche?
The true significance of the phrase, however, is its importance to Blanche's conception of herself, in terms of class, race, and gender. Blanche sees herself as a Southern belle—someone who should be cared for, tended to, who should be treated by men with gallantry. The phrase suggests that "strangers"—strange men—will perceive her upper-class status and react to her with respect. Implicit in this conception of self is her identity as a white woman and the privilege that goes along with that status. This image of herself is at odds with the reality of her situation and with her own agency. Blanche moves through the world pretending to be one thing, while the world (e.g., Stanley) sees her in quite a different way. While Stanley's assault may have triggered her mental collapse, Blanche has been living a kind of fantasy long before she came to stay with the Kowalskis.
There is a kind of tragic irony to Blanche's last words, "Whoever you are I have always depended on the kindness of strangers." She is at this point in the play being escorted out of Stella and Stanley's apartment by a doctor, who has come to take her to a psychiatric institution. The fact that she doesn't know who the doctor is here ("Whoever you are") points to how psychologically disorientated she is by the end of the play. And, tragically, she has been reduced to this condition in large part because strangers have not been kind to her. Mitch told her, "You're not clean enough to bring in the house with my mother," when he found out about how many men she had slept with. Stanley violated her when he raped her offstage. Even her own sister has betrayed her.
Blanche's final words are also, however, grounded in truth. Earlier in the play Blanche told Mitch that she "had many intimacies with strangers. After the death of Allan—intimacies with strangers was all I seemed able to fill my empty heart with." In other words, after her teenage love, Allan Grey, committed suicide, Blanche had sex with many men, strangers, because that was the only way she knew how to fill the emptiness, albeit temporarily, left by Allan. The reference to "strangers" in Blanche's final lines may remind an audience of this earlier reference to her many "intimacies with strangers." Arguably, Blanche's demise can be traced back to Allan's suicide, and her part in it. Her descent into madness is perhaps merely the tragically inevitable consequence of the absence left by Allan, and the futile, repeated attempts to fill that absence with "the kindness of strangers."
Toward the end of the play A Streetcar Named Desire, Blanche meets her tragic end. She is sexually violated by her sister's husband, Stanley. This was the culmination of a cycle of bullying, threats and harassment caused, in part, by Stanley's perverse want to control Blanche the way he controls his own wife.
Stanley also wishes to expose Blanche's past in order to diminish her and make her socially disappear for his own sick purposes. When he finally rapes her, she becomes a completely broken woman who loses her mind.
The line "Whoever you are, I have always depended on the kindness of strangers" is known as Blanche's "famous last words." She says those words to the doctor as he leads her away to the mental institution.
Essentially, what the words mean is that Blanche, once rich, popular and independent, has ended up in an abyss of disrepute, poverty, and debauchery. She switched her life's moral code, wasting way in meaningless sexual encounters for the sake of basic but temporary comforts: company, monetary benefit, attention, a need to feel wanted, and temporary satisfaction.
All of these ephemeral things depend entirely on "kindness"—sexual attention or physical attraction—from others. These two variables, however, are just like the things that Blanche gets from them: shallow and temporary.
Therefore, what Blanche is suggesting is that she is no longer in control of her own actions, nor of her life. Granted, she had lost control way before she actually lost her sanity. Still, then and now, she depends on that "kindness," "whoever" it comes from, whether it is the doctor leading her out the door, or Mitch, or any of the guys from her past.
These words are also powerful in that they completely disempower and disenfranchise Blanche. The irony is that, at all times during the play, Blanche attempts to present a facade of someone who is "put together" and in control of what she wants; someone who is even presumably planning for her future. To openly state that she depends on kindness renders her nothing short of a beggar, something that the true Blanche Dubois would have never allowed herself to be.