Water usually symbolizes emotions. A faucet controls the flow of water. Faucets turning on and off automatically as Largeman passes could be saying something about his emotions being controlled by forces outside of him (drugs his father has prescribed) or about the limited range of his emotions at the start of the movie - this happens in a very sterile-looking bathroom after all.
The scene might also be foreshadowing an awakening of emotion in the character, something he will cause by deliberate action: walking too close to the sensors = stopping medication (both cause the emotions to start to flow). There is another scene in his parents' house where he tries to stop a leaky faucet and fails - perhaps he wants to stop the flow of emotion that has begun.
There are other water references in the movie. There is a rainstorm and an ark (prepared for a flood). There is a swimming pool into which Largeman conspicuously fails to jump. The tub is a symbol - it can hold a lot of water. The quarry can hold even more. There are Sam's tears and eventually Andrew's.