As long as you put them to bed properly, no reason to be concerned letting it sit for months.
In fact starting once a month is likely doing damage. That's long enough for the oil to drain off all the internal surfaces. Until oil gets pumped back up to the surfaces, you are getting accelerated wear. I would prefer to keep that to a minimum.
Now, if you are going to exercise it, it needs a good load. Particularly the newer inverter generators may not get up to full operating temp and burn off any moisture introduced as they will run at a low idle with no load. A 1500w heater is a good option for a small generator.
Obviously, shore power needs to be disconnected unless the generator has a means of syncing with the shore power (small generators we are discussing won't)
