There are safes that are found during missions that require a certain combination in order to open them. There are clues somewhere in the level, either hidden nearby or gotten from doing a sidequest of some kind, that give out the combination for the safe. Each safe contains items that Corvo can use or give him a high number of coins.
The safe in Dishonored can be found in the sewers section of the mission, which is the second half after escaping from the prison. In the room before you climb up and overhear some guards talking about you, you will find the safe on the left side, along with a note as to the combination's whereabouts. The combination is located on the shelf behind the safe, and it is 451.
HIGH OVERSEER CAMPBELL
#1) This safe is located in Dr. Galvani's Offices, which is in the area past the first Wall of Light in the Distillery District. It's on the second floor, and you can read his journal in his bedroom, which mentions the "28th day of the 7th month." The combination is 287.
#2) There is a locked door in the Kennels section which requires a combination. You can get the numbers in a room, from a note, which mentions the Seven Strictures. The combination involves finding out the mentioned stricture names and determining which number they are. The combination is 217.
#3) This safe is located in the Back Yard area, and is inside the Bunkerhouse. You can get the combination in one of two ways. You can help a rogue Overseer and his sister escape from two Overseers, who are in the alley in Back Yard leading to Holgar Square. The other method of finding out this combination is by finding the note underneath the bed, next to the safe. The combination is 203.
HOUSE OF PLEASURE
The only safe in this mission is located inside of the Art Dealer's Apartment, on the third floor. In order to get the combination, you will need to torture the Art Dealer inside of The Golden Cat until he tells you it. The combination is randomly selected from one of the following every time you play the mission: 879, 138, 656, 679, or 696.
THE ROYAL PHYSICIAN
#1) When you enter the second area of the mission, you will find the safe on the third floor of the building directly in front of you. There is a clue on the table nearby, which mentions finding the numbers in three specific paintings. These are "the crowded place," the shipping yard and the whaling ship. The combination is 473.
#2) This is located in the last area of the mission. You will find some people trapped inside of a Wall of Light, so take out all the guards in the area, and disable the Wall of Light. Once you do that, the lady will tell you the combination. The safe is located behind a painting on the third floor, directly above where the Wall of Light above was. The combination is 294.
RETURN TO THE TOWER
The only safe in this mission is located in the Lord Regent's bedroom, on the second floor of the tower. You can get the combination from the broadcaster in the Broadcast Tower, which is a part of the nonlethal objective to eliminating the Lord Regent. The combination is 935.
THE FLOODED DISTRICT
#1) This safe is located in the second area of the mission, but the combination is found in the first area. When you get to the bridge from the chain you climb up in the beginning, face towards the Daud's base marker and look to your left to see a balcony. Get onto the rooftop, then onto the high one and drop down and get onto the balcony (Sokolov Painting in here). In the next building is the note that points out the combination (428).
To find the safe, head into the second area of the mission. You will be on a large wooden walkway that is over some water. Drop down into the water and look to your left for a small alley. Go through here and you should see some stairs leading down to a door. Go through the door and follow the path downstairs on the left to the safe, which is underwater.
#2) This safe is located in the sewers. When you first arrive at the gate leading deeper, go to your left and follow the red pipe. It is in the room on your left after breaking through the wood. The clue for the combination is on the desk to your right, which mentions the months. The combination is 528 and there's a Bone Charm in it.