Scattered around Dead Island 2 are safes and other items that require you to find a key to unlock them. Usually, these things hold some kind of reward such as weapons or blueprints.
One of these things is the Security Guard Cooler, which you will find as you come to Monarch Studios as part of the main story. While it may be difficult to understand where the key is, as it isn't very obvious at first, you won't have to go far in order to get your hands on the key.
The Key's Location
You have most likely come to Monarch Studios as part of the Michael Anders and the Holy Grail quest to find Michael for Emma. As you approach the studio, you will see the entry gate that is closed with a blown-out portion of the wall serving as a new entrance for you. On your left, you will see a security booth that you can unlock by destroying an electrical box on the outside. Do so, and the door will open, revealing the cooler on the ground and locked. When you try to interact with it, it will tell you that you have to find the Security Guard Cooler Key.
What you need to do is just follow the story quest for a bit. As you follow the steps, you will eventually enter Studio 7 where you will need to complete some puzzles and fight your way through the usual cast of zombies. Eventually, you will come out of the studio in a place called Action Alley with your mission marker telling you to go left. Don't go left, instead, go right to see a trailer for someone named Jimmy Montana.
Enter the trailer to find a zombie chowing down on what we can only assume used to be Jimmy Montana. Take this undead eater down, and they will drop the key for you. Now, you can exit the trailer, go to the right, unlock a gate that could only be opened on this side, and head back to the security booth in order to open the cooler and claim your rewards.