Would you please criticize "Rappaccini's Daughter" from a Structuralist approach?
Of course, the key to answering this question effectively is an accurate understanding of "Structuralist." This word is used differently by various people and in various time periods, so there is probably no one definitive application of this word in literary criticism.
I think there is one Structuralist concept that may be more useful than others when analyzing "Rappaccini's Daughter," but please understand that someone else (namely your teacher or professor) may have something else in mind.
Structuralism is generally concerned with how a story is framed or built and then with how that structure conveys meaning. Most stories are constructed like something else that has been written--there are very few new forms but many variations of familiar structures. (Think of Shakespeare's Romeo and Juliet and West Side Story, which are essentially the same in plot, character, and theme.) These familiar literary forms include symbols, myths, and archetypes, so let's take a quick look at each as they're used in Hawthorne's story.
Symbol: Clearly the dominant--and dangerous--purple plant is the centerpiece of Rappaccini's garden. It is both beautiful and deadly. The consistent references to this plant being Beatrice's sister, their matching purple attire, and the later knowledge that this plant was actually "born" on the same day as Beatrice tell us the plant is a symbol of the girl. The plant is deadly to all but Beatrice, and we can't miss the fact that the same must be true for the innocent girl. This kind of parallel (something in nature representing something human) is common throughout literary history.
Myth: In Hawthorne's time, the name Beatrice would have been associated most with two women in history. One is a character found in Dante's Divine Comedy. She was a symbol of purity and love, guiding him to heaven. The other, Beatrice Cenci, was an innocent girl who was rather unwittingly used by her family to kill her father but was then killed for doing so. These two sides of Beatrice were clearly pre-existing myths which Hawthorne utilized in this work.
Archetype: This one is fairly straightforward and applies primarily to Rappaccini himself. He is a dispassionate manipulator of man (or in this case woman--his daughter) to promote his own ends. This is a common pattern throughout literary history. The most obvious archetype for this story is Dr. Frankenstein and his monster/creation.
In short, the structure (symbols, myths and archetypes) of this story is one which unfolds as we read it yet is not unexpected or unfamiliar.