A dynamic character is a character who grows and changes over the action of the text. The opposing character is called a static character. This character does not change or grow over the movement of the text.
In regards to the protagonist in Gilman's short story "The Yellow Wallpaper" the protagonist does, in fact, change throughout the text.
The story opens detailing a move to the country where the narrator is supposed to be getting better. While it is never stated, the protagonist has been assumed to be suffering from Post-Partum Depression. Therefore, she is already suffering from a mental illness.
During her stay at the country home, the protagonist grows rapidly more and more unstable. In the end, she has gone completely insane.
Therefore, based upon this fact alone, the character does change over the course of the story. But, one could argue that she comes into the story mentally unbalanced and ends the story still mentally unbalanced. In regards to this interpretation, the protagonist would be considered to be a static character.
The denotation of either static or dynamic simply depends upon ones own personal interpretation of the changes which takes place.
As for the other characters in the story (Jennie and John) readers do not know enough about them to deem them static or dynamic.