The above answer is quite accurate, but Walton's Romantic nature is revealed in other ways as well. Frankenstein is not revealed as Walton's "nemesis" but rather as the one friend Walton finds on the the sea. In his letters, it is clear that Walton is quite lonely. He tells his sister that the one thing missing in his life is a friend, a companion with whom he can share his achievements. Of course, one might think that Walton would sympathize with the creature in such a case, as the creature suffers from crippling loneliness as well. Instead, Walton finds sympathy for the man Frankenstein, & is just as repulsed by the creature as everyone else in the tale. Yet the captain is still Romantic by nature; even his recollection of the first mate's failed love affair reflects a Romantic world view.
Also, Walton reveals his Romantic nature by his attachment to art. He states that in his youth he desired to be a poet, which could be considered a reference to Shelley's husband and their circle of Romantic poet colleagues. Yet the sea continued to call to Walton, & so he chose the life of an explorer, which, as the previous poster mentioned, is Romantic in itself. Finally, he liberally quotes from "The Rime of the Ancient Mariner." You wouldn't exactly expect a hard-bitten sailor to spout off poetry, but Walton is not your typical sailor. Coleridge too was a Romantic poet, & a friend of Shelley's father. Hearing him recite that poem when she was 8 may have greatly inspired Frankenstein. Whether it did or not, it's clear that her Romantic influences helped create thee character of Walton.
Robert Walton is a romantic, in the sense that he is idealistic and believes that imagination reins above all else. Basically we can see by his actions and his "profession" or vocation that he is a romantic. His unrealistic vision of finding a passage which would unite the two oceans--at that time in history--was indeed a romanitc notion. In the end it is clear that he found his nemesis in Frankenstein, who too was too idealistic.
Robert Walton is sort of a narrator, through his letters to his sister, Margaret Saville. He tells the story of Frankenstein. He is an adventurous man who is overcome with unbridled passion to acheive his goal. His goal was to explore the North Pole. He wanted to be the first to find a passage through the artic, which would reveal a connection between the Atlantic and Pacific Oceans. His ship becomes trapped on the ice and he thinks he sees Frankenstein.(He is very lonely and needs a friend.) He says that they took Frankenstein on board their ship to nurse him to health. Frankenstein tells his secret story to Walton and he tells Walton how his aspirations led to his demise. Supposedly Walton was the last to see Frankenstein alive.