Multimeter on volts setting (20V range if you have a manual range meter). Poke the black probe at the ground wire on JAMMA (should be black) right at the PCB edge. There may be marked test points. Use those if available. Red wire on +5 (usually red) for testing 5V and +12 (usually orange in arcade cabinets but sometimes yellow) for testing 12V.
5V should be 4.75-5.25V. Obviously, the closer to 5.0V, the better. 12V should be 11.25-12.5V or thereabouts. Again, closer to 12.0V is better. Most arcade power supplies have a knob on them to adjust voltage. Try to get things as close to 5.0V as possible. 12V is less picky.
Of course, these are all just DC measurements. There are some things that can (somewhat rarely) happen to power supplies that cause them to read OK on a DC meter but actually not be good. Without more sophisticated test gear (true RMS meter or o'scope, mostly), it's difficult to test for these conditions. Usually one just replaces the power supply if this is suspect. A power supply is usually only about $25 shipped, so that's not too bad.
The other thing that can sometimes cause this failure is overheating. The stock kit (which you should have in your dedicated cabinet) has a fan. It's usually directed to cool the speaker amps and blows a little air over the GPU. You can redirect it to cool the CPU and GPU instead if you want. In most home setups, the volume isn't loud enough on the cabinet for the amps to get very warm.