A generator can be run in the rain just fine.
Although I would recommend putting the generator underneath something such as a portable canopy or under a carport etc just help keep the rain off of it.
But even then the generator will run okay in the rain although you risk getting moisture in the fuel through the fuel tank lid and condensation.
When I'm at home I keep my generator under my carport when I'm running it to keep it out of the rain directly and being under the carport allows it to vent properly as well so there's no obstructions as there would be in an entirely closed off or nearly closed off building etc.
Never run your generator indoors though as it can build up deadly carbon monoxide in a short time and kill you.
You could also place the generator inside a detached garage with the garage door open and the exhaust venting outwards or place it in an enclosed trailer with the back door open.