Since normally nothing changes in an analogue circuit on its own, except for defective electrolytic capacitors, faulty mechanical switching contacts and damage due to moisture, heat, force or contamination, I would first start by examining the battery compartment. Corrosion caused by leaking battery electrolyte also continues behind the battery contacts and attacks the circuit behind them, even if everything looks fine from the outside.
The Nikon Motor Drives that I have opened so far are all tightly wired, which means that you should be careful when taking them apart and assembling them.
The solid housings can often only be opened with patience.
I would definitely get the service manual and proceed systematically when troubleshooting by visually inspecting the circuit and using a multimeter basing on the analysis of possible errors. Test points are also listed on the page above making troubleshooting more effective.
When I was assembling an MD-12, I had pinched the positive connection of the battery cable. There was a short circuit to ground (metal housing) and the batteries quickly became hot due to the short.
Perhaps the cause is immediately apparent; I would take a close look at the wiring in the housing. If a component has burned out, there is a cause that must be rectified before it can be replaced.
A friend had asked me to look at his Nikon MD-12 Motor Drive. The Motor Drive for the Nikon FE(2) and FM(2) hasn't been available for a long time. A contact in the battery compartment was corroded but was poorly cleaned. I didn't say yes right away because I usually only work on my own stuff...
As always:
All information provided without guarantee and use at your own risk.