A few things to check out.
1- Most builders do not run both the positive and negative cables all the way back to the batteries. They run the positive back to the battery or solenoid or buss bar. The negative cable is bolted to the body frame work or chassis frame. Most likely not getting a good connection or has corrosion at on of these points. The ground connection on the generator is the lower case or housing, and not the best of spots. Much better going right back to the engine of the generator.
2- Just because a battery shows 12 or 13 volts does not mean it is good. Have seen way to many big batteries that are made for starting large diesel engines show 13.4 volts and would not even start a lawn mower.
3- If the batteries were allowed to sit dead for even a few days and are not charged they have most likely sulfated and are shot.
Start at the batteries and work out to the generator checking for voltage and the ability to carry a load to crank the unit.