First I will observe that one gets a lot of strange question on the Internet. An influential monster movie seems almost to be a contradiction in terms. The vast majority of monster movies may be entertaining in themselves, but they are hardly films that are considered influential. Okay, here is my list. Between World Wars Germany made a number of fantastic films and for one time in film history the great artistic films being made were fantasy.
Expressionism is the distortion of the visuals to present an emotional reality.
The Expressionist movement used distorting shadows, weird camera angles, and often odd, angular sets to create emotional effects. Using the techniques of German Expressionism, and not a little talent from European refugees, Universal Pictures started its own cycle of horror films. The score by Max Steiner was his first great score and made a name for him. Producer Val Lewton was something of a maverick and made a series of intelligent and literate horror films. This film, full of sexual double-meaning, was the first set in familiar, identifiable surroundings.