Ok - checking stuff out for dummies, chapter 1

This applies to the generic cabinets, not the dedicated stuff.
1) Do not power anything up yet. Never fit or remove game components with the power on. Power switch is usually at the back, it's safer to actually unplug them.
2) Hunt for the keys - there's actually a pretty decent chance that if there's a small office in there, you'll find a whole tub of keys. They're flat rectangular looking things as standard for most of those cabs.
3) No keys? You'll have to drill the locks. It's difficult, might find examples on Youtube - you need a pretty heavy duty drill and toughened bits. Even then I've ended up with shards in my leg from an exploding bit before now. Alternative: hire a locksmith to open them. The door you care about most is the largest one on the front.
4) Inside will be a circuit board (if they were left inside) - these vary from about 1 foot square upwards, sometimes multiple layer. You'll see an edge connector with a 56 pin harness on it, remove that. If you see a large plastic looking thing instead with PC style connectors, that's a JVS hookup (probably Naomi or similar), disconnect those instead. Often the boards in those cabs are mounted on wooden panels which slide out. Pay particular attention to which way up the connector is, standard 56 pin JAMMA pinout will have a key position on pin 7 - pins 1/2 are ground and 3/4 are 5v. Familiarise yourself with the pin locations on the connector. Google for 'JAMMA pinout'.
5) When all the boards are removed, as Starcreator says take good photos of them, just blow loose dust off unless you have an antistatic brush. Do not try to clean them. These are sensitive to static by the way, and physically fragile.
6) When it comes to testing the cabs there's no easy way past 'suck it and see'. You will need a multimeter and have identified a game board which is nearly worthless in the step above. First you power them up with no board installed - this is to get any potential "I've been turned off for years and plan to explode" situations out of the way first. Do this in a dark room and you should see the screen being 'not dead' but still black. If you did find the keys you can look in the back and see if the tube neck glows, it's the glass bit with a small pcb on the end. Don't ever touch the back of the screen, you're not qualified. You'll probably hear a small amount of 'popping' initially, if this calms down then you're good.
7) Multimeter time - DC volts, measure pins 1+3 on the 56 pin connector with no game board installed and power on, expect to see around 5v, if it's anything 5.8 or above you need to adjust the 5v supply - ask about that if you need to later. These run at 5v and higher is dangerous, but we expect them to be a bit higher since there's no drain with no board installed. There are no dangerous voltages on that harness, but don't go shorting it out.
8) Any that have a live screen and working 5v, plug in a test board. If you've supplied good photos to collectors to help with ID, we can probably pick out one which is almost worthless, and thus perfect for testing. Turn the game on and see what happens
Or you can roll the dice and be brave

I wouldn't. If a screen blows it's not impossible for voltages to feed back down through the video ground damaging the game board. I've seen it happen, additionally old power supplies can misbehave and you don't want to cook games.