Once you determine which side is leaking turn off both supply valves.
How to fix a leaking wall mounted kitchen faucet.
Turn the packing nut counterclockwise with a wrench to remove it.
If a faucet has compression valves replace the washers on the valve stems.
More often than not a a dripping faucet is due to it s wear and tear.
You have to disassemble the faucet so turn off the shutoff valves.
Set this aside also.
There are few problems with wall mounted faucets other than occasional leaks which are easy to fix.
If your valves are stuck shut off the water main.
Turn the valve stem you can use the faucet handle this to open the tap.
There is constant friction because of which it wears out.
If the faucet has cartridges replace the cartridges.
Determine which side of your faucet is leaking by shutting off the water supply valves one at a time.
Over a period of time with every use the washer is forced against the valve seat.
How to fix a leaky wall mount kitchen faucet remove the screw that holds the handle of the faucet stem.