There are several similarities in the Old Testament story of the Garden of Eden and Nathaniel Hawthorne's "Rappaccini's Daughter."
One line in the short story that makes me think of the story of the Garden of Eden is Hawthorne's description when Giovanni first sees the garden in the Rappaccini's yard:
...it might once have been the pleasure-place of an opulent family...
If we read this line quite literally, we can easily see that Adam and Eve were the richest people on earth, not just because they were the first, but because they had all they could ever have wanted in the garden, with command over the animals—wanting for nothing.
Of course, there is also the presence of the serpent in the garden, the creature that tempted Eve to commit the first sin. Hawthorne also mentions a serpent-like presence in his garden:
Some [plants] were placed in urns, rich with old carving, and others in common garden-pots; some crept serpent-like along the ground...
The reference to an evil presence in this backyard garden—much like the evil in the Garden of Eden—is mentioned when Giovanni first sees Rappaccini:
...for the man's demeanor was that of one walking among malignant influences, such as savage beasts, or deadly snakes...
Hawthorne makes a direct comparison himself to the Garden of Eden:
Was this garden, then, the Eden of the present world?--and this man, with such a perception of harm in what his own hands caused to grow, was he the Adam?
Another allusion to the Garden of Eden and its occupants comes from a description of Beatrice Rappaccini:
Evidently her experience of life had been confined within the limits of that garden.
Here Beatrice is presented to be an Eve-like figure, only knowing life in the Garden and nowhere else (before eating of the fruit).
Hawthorne again draws a comparison between the two gardens:
And down he hastened into that Eden of poisonous flowers.
Overall, the most obvious parallel I see between the two stories is that there is evil lingering in the midst of the beautiful plants and flowers of both of the gardens. Also, the females (Beatrice in Hawthorne's story and Eve from the Bible) are dangerous figures to the men in the garden with them.