I'd start at the converter. Do you get 12 VDC out of the converter? If not do you have 120 VAC nominal at the input to the converter checked with a meter? If yes you have a converter problem. If no you have an AC power problem and probably low batteries.
If it's a converter problem check all the fuses and built in breakers. If they are OK you are into the pull it out and send it in step.
If it's an AC problem work back through that feed chain.
FWIW - measuring 12 power the only thing you have to worry about is shorting it to ground. Otherwise it won't hurt. Park rings, watches and similar metal somewhere else while you work.
120 VAC can hurt or kill you. If you are not comfortable messing with it call a pro or get some test leads with alligator clips. Pull the plug, shut off the generator and kill the inverter. Make your test connections with the clip leads and then plug the unit back in. If you need to you can slip a short piece of stiff wire into any screwed connection to make a test point. Make sure the power is off when you are doing it. One of the best tools you can have is a non contact voltage tester like this:
Ideal Volt-Sensor Non-Contact Voltage Tester-61-063 at The Home Depot
That one is not my favorite because I prefer penlight batteries but it is cheap and should work. Anything like it is fine.
Always test it first by touching or coming close to a known live wire. If it lights keep your hands out of there. ;-) Once you have a feel for it you can probe around safely as they do not conduct power. The problem is that they do not tell you how much voltage other than too much to stick you hand into. That is why you test with that and connect the meter when the light does not come on.
Come back with some answers and folks here can give you next steps.