I’ll start by asking how long you expect to store it? But presuming around 6 months, and presuming the genny runs off of the main gas tank, I’d run the tank low, preferably with non ethanol in it, add a stabilizer to what’s left in the tank, run everything for a bit, run the generator out of fuel (assuming carb generator) shut it down and not touch it until spring.
Or you can do the start it every month thing which is really only beneficial to the genny if you leave it full of fuel that can degrade and evaporate. Even then, I wouldn’t bother starting the moho engine. However if you must go through that exercise, which is mostly a placebo, then run it long enough to get everything heat soaked. Ice cold engine will make some moisture as it thaws out after a long sleep.
Many will say to fill the fuel tank. That practice is of no benefit IMO on modern fuel tanks/systems. I winterize or summer-ize or just plain don’t use regularly, around 15 engines a year at the house. Small engines with carbs either get run out of gas or left with AV gas in them. Small efi engines get shut down with AV gas or stabilized gas and don’t get started until needed again.
Haven’t intentionally filled a fuel tank before storage since about 1995 when I figured out that it doesn’t matter on anything that doesn’t have a metal tank. Only issue is old fuel going bad, which is much harder to rectify if you have a full tank vs an empty tank.
Another tip, ethanol blend typical fuel is great to top off a low tank that’s been sitting. The ethanol will absorb any normal moisture that may be present. Just like adding a can of HEET in the winter if you have fuel line freezing issues.
Lastly, one is all gung Ho the first year or 2 to “go run it every month”. What about 5 years from now? What if you’re not around for 3 months, etc.
just get it winter proofed and come back in the spring. Easier, cheaper, less time consuming and no downsides if you do it right.