The question as to whether Edna is a fallen woman would depend on two things: the era in which the story was written, and the era in which it is being read now. Both have very different expectations (if the tabloids and Hollywood "entertainment shows" are any judge).
In the time in which "The Awakening" is set, society's expectations were very different—at least for women.
Dr. Susan E. Ward, Professor at St. Lawrence University, notes that in the Creole society in which Edna lives, there are clear cut rules regarding what is expected of its women:
Women are to be pious and pure, obedient and domestic before they are to be anything else.
Edna's Victorian husband, Léonce, seems a proper husband who seeks out medical advice from a friend regarding Edna's behavior, and he even tries to make her feel better by allowing "new fixtures for the library." He is what society has made him, but not a bad husband at all. However, Edna is unable to be the woman society wants her to be: she is unhappy and unfulfilled. Unlike Adèle Ratignole, who is the perfect wife and "mother-woman," Edna's awakening is, in part, coming to the knowledge that she is not meant to be a wife and mother.
The rest of the changes Edna goes through deal with her romantic connection to Robert Lebrun, who she becomes infatuated—perhaps in love—with, though their relationship is never "consummated:" he and she do not run away; he tries his best to do what is right by this married woman. The loss of Robert devastates Edna. The third part of Edna's awakening comes at the hands of Alcée Arobin. Their relationship is a physical one: Edna is awakened to an exciting and passionate side of herself.
However, none of these things brings Edna inner-satisfaction. She becomes more and more despondent over what she wants her life to be, and what it actually is. Beside the emotional and physical ramifications of her marriage and her infidelity, is Edna's desire to exist on her own without her husband's financial support as an artist; she does not even have a healthy relationship with her children, in her heart. If anything, society would expect her to put her children before everything else, something Edna feels she could not do.
Professor Ward also notes:
Perhaps, had Edna Pontellier lived in a different time and place, she might have acted differently.
I would suggest that in a different time and place, Edna would be perceived differently. However, in the context of her time, I believe she would have been perceived as a "fallen woman," defined as:
[a word used in the past for] a woman who had sex with a man she was not married to
It is important to note that the definition mentions that the term is one "used in the past." Today the idea of a "fallen woman" is not perceived as it used to be—except perhaps by some more conservative people. In a Victoria time-frame, Edna was a "fallen woman," as she has an extramarital affair with Alcée.
Society at the time Chopin was writing had strong opinions about her work:
Even though women's roles in the real world were changing, Chopin's frank treatment of female sexuality, social impropriety, and personal freedom shook the literary world.
Kate's books were banned, and her work criticized. Today, her stories would barely raise an eyebrow, but during her life, Edna was indeed "fallen."
Chopin, Kate. The Awakening & Other Stories. Hertfordshire: Wordsworth Editions Limited, 1995.