There are 3 ways to open Safes in Far Cry New Dawn:

Remote Explosives
Lock Picking Perk
Repair Torch Perk

Remote Explosives
You can simply throw a remote explosive at a safe to blast it open. It doesn't have to be perfectly placed, just needs to be close to the safe. You can craft remote explosives in Throwables section of the weapon wheel. Open the weapon wheel and press D-Pad Up / D-Pad Down to scroll to the remote explosives. It requires the ingredients black powder and blasting cap which you get from looting dead enemies.

Lock Picking Perk
This is the most elegant way to do it. Buy the Lock Picking Perk and hold Square (PS4) / X (Xbox One) to open the safe quietly.

Repair Torch Perk
The repair torch can repair vehicles but is also good for opening safes. It makes the Lock Picking Perk somewhat irrelevant. Buy this perk and select the repair torch from the weapon wheel to crack open any safe.