Instructions
Travel to the Trouble Center in Rogueport, which on the east side just before the bridge.
Press the "A" button next to the listings inside the Trouble Center to view them. Select "Need a Key!" to view Garth's request to find his key and accept it.
Head over to Garth, who is standing outside his house on the other side of the bridge. Talk to him to learn that he dropped his key in the badge shop.
Head over to the inn, which is the building with a large bed sign on it. Go upstairs and out the door on the west side. Follow the path to the rooftop of the badge shop, where you will see Garth's lost key. Walk over it to pick it up automatically.
Head back to where Garth is standing and talk to him. Agree to give his key back and Garth will unlock his door.