Mary Shelley herself, as with many of her family and friends, saw a need for social change, but that is not really the main theme of Frankenstein. In the novel, science is seen as out of control, leading people to value their own intellectual pursuits and quests for discovery and innovation at a level that distorts their moral judgments. Victor's own quest for knowledge and the power to create life lead to disaster for innocent people around him. The innocent Justine, for example, is murdered for a crime for which Victor is, at least indirectly, responsible.
One area where one could say that Shelley advocates for social change is in her discussion of how the monster feels that positive and negative influences have shaped its life. It emphasizes the importance of love and good role models as a key to child rearing, perhaps an indirect critique of the harshness of English public schools. This is a difficult argument to support, though, because Victor is raised by kind and loving parents but still turns out self-centered and destructive.
Shelley seems to be warning us that the (in her time radical) Enlightenment emphasis on individualism and personal freedom needed to be tempered by a greater degree of empathy and social awareness and that if science reduced the influence of religious morality, it needed some other system of morals and ethics with which to replace it, to avoid the sort of negative consequences seen in the story. Therefore, the monster represents the power of science unconstrained by moral training, something Shelley saw as a future that needed to be averted.
The Romantic movement, like other new artistic movements, was a reaction to the previous period (the Industrial Revolution and Age of Enlightenment). Given their dissatisfaction with what preceded them, the Romantics elevated the individual, imagination, and nature.
All this said, Mary Shelley's Frankenstein can be seen to act as a metaphor for her desire for social change. Given the influence her parents (William Goodwin and Mary Wollstonecraft) had on different movements--especially the feminist movement--the lack of feminine power in the novel speaks to Shelley's own call for change particulary given that Victor's views of women and their power are not positive. He tells Elizabeth that he will only marry her if she waits to hear his secret. Also, Victor holds no qualms about destroying the female creature. By far, the most poignant example of Victor's views regarding women is shown by his taking away of the one thing woman had over men--the ability to give life.
Another example of Shelley's call for change could come from the warning the novel provides. While scientific advancement is important, some things should simply be left as they are (illuminated through Victor's search for Forbidden Knowledge). Therefore, while Shelley may have agreed with scientific advancements, the novel metaphorically serves as a warning against going too far.