Quite early on in your time playing Fallout 76 Wastelanders, you should come across Gauley Mine. You will travel through the area while working on the questline to find the Overseer. Keep an eye out for Slocum’s Joe and Arktos Pharma, and you will know that you are in the area where you can find Gauley Mine.
You don’t have much reason to venture into the mine, but there is a locked room filled with some loot in there. If you want to look around to find the code, you will need to look for a side tunnel, then follow it until you come to the catwalk with some lockers nearby. The code is in the lockers.
It is much easier to tell you the code, though, and it’s 071990. This will open up the supply stock for you, and you can get your hands on some much-needed loot.
To find the cage itself, venture further into the mines, and it will be waiting for you. Type the code into the keypad, and you will be good to go.
You will need to be careful down in the mines, as Scorched enemies inhabit them, but they are not too tough and are really just an extra source of loot. It is a good idea to keep an eye out for other players too, as this is one of the earliest points in the game’s campaign, it is not unusual to run into people working on the same quest line as you.