It is impossible (unless Stevenson left notes to the effect) to know if Mary Shelley's Frankenstein had a direct influence on his writing of The Strange Case of Dr. Jekyll and Mr. Hyde.
In terms of the social components of the time, Shelley was greatly affected by the Romantic movement, growing up in the company of writers and philosophers of that time, and even marrying Percy Bysshe Shelley, one of the most noted of the Romantic writers. Romantic writers extolled nature and supported the plight of the weak. Stevenson was writing during the Victorian era, where conflicting views of Christian morality and hedonism waged war throughout English society. Utilitarianism was focused on the greatest good for the largest number. This "school of thought" concentrated on seeking out pleasure foremost, and avoiding pain whenever possible. Evangelicalism was much more rigid and Church-centered—moving far to the other extreme— searching "for every moral blemish." However, it also is credited for many of the humanitarian efforts this time period is known for.
Robert Louis Stevenson's characters of Jekyll and Hyde seem to clearly represent these two extremes. Hyde is the malevolent side of the scientist, seeking to find pleasure where he may and resisting any thing or person that interferes with his happiness—hence his horrific temper. Jekyll, on the other hand, is the moral side of the "creature" his science has created—it tears him apart physically, morally and philosophically.
In Frankenstein, we see more an internal struggle of the scientist (Victor Frankenstein) with society and religion at large. He "plays" at being like God in creating life (and ignoring his ethical responsibilities), and then ignores his creation, giving no attention to his moral obligations to the creature he brings into being.
The themes of Frankenstein and Dr. Jekyll and Mr. Hyde seem similar. Ironically, Victor Frankenstein seems to have multiple personalities: the scientist who is willing to risk all to realize his goals, and later a much more conservative man who regrets his immoral behavior. The two sides of Dr. Jekyll also present a sense of "split personalities." It is with this thematic element that the two stories seem to have the most in common.
…Frankenstein's creation has often been characterized as [Victor's] "dark side…"
Though Stevenson wouldn't write his tale for close to seventy years after Shelley's, its possible to see a strong similarity between Victor, and Jekyll and Hyde.
The Frankenstein monster could be considered a model for Stevenson’s Mr. Hyde, the embodiment of all of Henry Jekyll’s less “proper” thoughts and actions.
Other similarities include Victor's sense of alienation from family and society, mirroring the creature's sense of isolation. Jekyll and Hyde are also unable to connect with society: Jekyll because of his actions and loss of control over his alter ego, and Hyde because of his inability to fit in with civilized society.
Other themes that apply to both stories are "science vs nature" "duty and responsibility," "appearances vs reality," "identity," "supernaturalism," "change and transformation" (which applies more to Victor than the creature," and "good and evil."
While it is hard to say if Shelley's story directly affected Stevenson's creation, they deal with very similar thematic elements.