To expand on this: Turbogenerators use hydrogen because it causes little drag, its cooling properties are good for a gas and it's much cheaper than helium.

Liquid cooling would be better but cooling pipes are not very amenable to spinning around at 3000 RPM and you have to keep the rotor inside the generator as compact as possible so centrifugal force won't rip things apart.

The non-rotating part of the generator is typically water cooled in bigger generators by piping the water straight through the hollow copper windings themselves (in slower rotating generator types, water cooling is also used for the spinning rotor).