Well for one you need a tool, a multi meter. There is a function called a continuity test which is used to test if a piece of metal can carry a current to where it's supposed to be. You use that tool, by placing one probe on a known ground of your board and the other probe at the pin of the connector and if current can actually flow from one prove to the other, the meter will display on screen and/or beep telling you current is indeed flowing through.

By placing one probe on the ground plane and the other on the pin of the connector, you are looking to see if that pin is indeed connected to the ground plane. The MPU board is monitoring that pin and expects to find that pin connected to ground for it to continue.

Beyond that, to really understand more you need to start reading on the various guides on pinballs, or specifically basic electric principles either here on elsewhere on the net to really learn more. The first thing to really grasp is that metal carries electricity and a continuity tester is meant to test if this metal (wire, switch or other similar device) is able to carry that electricity by sending a voltage in one probe and expecting to see that voltage, or at least part of it, return to the other probe.

It could be possible, like you said, that switch in your coin door is broken, and even when closed the electricity cannot flow from one pin to the other (you test the continuity between both sides of a switch) and so the board doesn't find the ground it is looking for and is halting the boot sequence as a protection not to send high voltage to the playfield, because the door is open (and someone might have his hands in there).

Get a multimeter, read up on basic electricity, testing wires and switches and then you'll be one step closer to fixing your machine.

Once you can test whether or not those pins are grounded then you can figure out which side to test further. If the pins are grounded correctly , then the board as issues (maybe it also needs something else that it isn't mentioning). If the pins aren't grounded, then you have to investigate further on why they aren't when they should be. Could be as silly a switch with a wire connected to the wrong pin. Could be a severed wire beyond the switch of the coin door. It could be that even though you close the coin door, the switch isn't properly pressed in. It could also be that you didn't close the coin door and I wrote all this for nothing?

Get a meter! If this machine is yours, you will need that tool often.