Assuming you're using a multimeter, there will be one setting on the dial for testing continuity.
It will look like the picture in this link:
https://learn.sparkfun.com/tutorials/how-to-use-a-multimeter/continuity
(and a detailed explanation there).
Touching one lead to any ground point and the other lead to any other ground or grounded point on the cabinet, whether harness or just a grounded part of the cabinet, should produce a beep. That confirms it is continuous. Touching one lead to one of the 5 volt pins on the JAMMA connector, and the other lead to another 5V pin on the JAMMA connector should produce a beep. Those should be continuous. 5V, or 12V, or -5V (if your cab has that) should not have continuity to ground.
There are ground pins on the JAMMA connector, the video chassis should be grounded, the coin door and other metal components should all be grounded. Anywhere you see a small green wire terminated at a bolt or a screw on a metal part, it is grounded (and again, all metal parts should be grounded...but that's more of an exercise in equipment grounding, and we are mostly concerned with any shorts that may be causing your blank screen).
Do your continuity testing with the power off. The rest of the info at that link echoes what I've just said, as well as giving some other good steps for checking voltages.