What is the position of the writer? Is she objective or subjective ("The Story of an Hour")?Kate Chopin's "The Story of an Hour"

Your question concerning Chopin's "The Story of an Hour" deals with the narrator's tone and manner of presenting the story. 

Objective, as you use it, refers to the point of view of the narrator.  An objective narrator relates only the details of the story without interpretation or judgment.  Fiction can be objective, but this story is not.

The narrator's attitude toward the protagonist in the story is sympathetic, and she interprets her character with sympathy.  This is subjective.

Mrs. Mallard is presented in a positive light.  If the work were objective, she would be presented in a neutral light.  For example, the character is "young, with a fair, calm face, whose lines bespoke repression and even a certain strength."  "Repression" is an interpretation and judgment of the lines on the woman's face, and indicates that as a wife--even as a wife to a relatively good man--she suffers repression.  And "strength" is an interpretation, also, as well as a judgment.

Interpretation and judgment on the part of a narrator demonstrate that a story is narrated in a subjective manner.

Notice that I have interpreted your use of "writer" to mean "narrator."  We can't, strictly speaking, pretend to know what is in a writer's mind at the time of writing.  We try to refer to the narrator, rather than the writer.  At the same time, looking at this story and other works by Chopin, you are probably safe in saying that Chopin, too, is subjective in her approach to a woman's place in marriage and society.  She is a strong feminist writer, and often reveals the claustrophobic-like roles women are forced to play in society.