I would check the battery voltage right on the battery posts when trying to start the genny. If the voltage drops a way down then the battery is the problem. In that case I would check to see if the battery is being charged. If the battery voltage doesn't drop then the problem is in the wiring, most likely a bad connection.
Starting the genny takes a lot of current and it doesn't take much of a bad connection to create a problem.
Have you been doing the required battery maintenance, checking water levels and adding distilled water when required?
