I've never fixed a faucet on my trailer, but I have fixed one in my house. My guess is that the leak is at the point where the cold water enters the faucet housing.
Before disassembling the faucet, get a thing called a basin wrench, which makes it possible for you to reach up from underneath and tighten that connection.
This is what the tool looks like -- the gripper at the end rotates on its "wrist" to wrap around the nut under the faucet:
Good luck! And hopefully someone with first-hand experience with a trailer faucet will chime in and let us know whether the repair is similar to those in an ordinary house.