To unlock the Safekeeping trophy in Far Cry New Dawn, you will first need to purchase the Repair Torch for 3 perk points or Lock Picking for 6 perk points. Once you have purchased either of them, you will now have the ability to open safes.
Safes are black boxes that will be commonly found when you follow treasure hunt missions and can also be found in the open-world.
Once you find one, hold the Square button to open them.