With basically no load, you should see very nearly the same voltage at the generator as at the battery. A bad ground connection would be a very good place to start. Bad connections in general at and around the battery would be next in line.
Note that the generator starter is, in general, the most demanding instantaneous load you'll place on the house battery unless you have a fairly large inverter. Mine takes around 90-100A when cranking. It doesn't take too much of a poor connection to have a significant voltage drop at those currents.
