I assume you've checked the circuit breakers in the RV's load center. Either way, that's where I would start. I would use a voltmeter to verify there is 110V at the output of the appropriate breaker in the load center and work from there. I didn't think most RVs used a GFCI protected outlet for the converter, but since its mounted in the basement if its in an accessible location, they might.
I also believe that the converter outlet is usually a dedicated outlet, but again that could vary by manufacturer. Assuming there's 110V at the circuit breaker, can you see the wiring path from the load center to the outlet the converter is plugged into? There could be a hidden GFCI somewhere, a broken wire, or there could be a bad connection somewhere.