How does Frankenstien have elements of Romanticism?

Shelley's decision to present the novel in the first person allows her to dive into the emotions of her protagonist (Victor). This provides the opportunity to give the reader a romantic vision of each situation, setting, and character. Victor tells the reader how it felt to travel through Switzerland, rather than other novels that might illuminate each location with more description than personal perspective. Also, Victor constantly asserts the horror and misery of his internal struggle (a prominent theme in romanticism), which, in turn, allows the novel to create a dramatic contrast to the joyful moments in Victor's life.
The addition of multiple points of view (particularly that of Frankenstein's Monster) demonstrates the romantic tendency to prioritise the feelings encouraged by life and conflict. The hatred of that wretched fiend (Frankenstein's Monster), the sorrow of Victor Frankenstein and the confusion of Robert Walton blend into a potion that exemplifies the range of the human experience, a very romantic endeavour.

Shelley borrows from, and arguably critiques, the Romantic tradition in this novel in a number of ways. In the first instance, we might look at the character of Victor himself. Like the speakers in Romantic poetry, his whole life is a struggle towards the sublime and a rejection of the boundaries that might be imposed upon him by the rational. The goal of making the creature represents his ultimate desire to transcend the mundane, moving beyond what is thought possible; remember Edmund Burke's note that "terror is in all cases . . . the ruling principle of the sublime." In achieving the horrific, the monstrous, the impossible, Victor epitomizes the quest for the sublime and the rejection of the rational.
On a more straightforward note, we can see obvious reflections in Shelley's novel of the Romantic preoccupation with nature, particularly with natural settings reflecting the action of the novel or feelings of the characters (pathetic fallacy). Consider that the novel both begins and ends in cold and frozen waters, contrasted against the inferno in which the Creature describes the ending of his life. Consider also the novel's preoccupation with fire and Shelley's lush descriptions of the cities Victor visits, particularly in Switzerland.