To check the converter, measure the battery voltage with the converter off, either by unplugging or turning the circuit breaker for the converter off.
Then turn the converter back on and measure the voltage. It should be a volt or 2 more, depending on your converter. If the voltage doesn't change the voltage it should be getting from the converter is not getting there. It could be a disconnected wire, a fuse or the converter could be bad.
It might help to tell us the make and model of the converter.
It might also help to check the ac voltage. The AC doesn't use the 12v except possibly for the thermostat.