I think October is an ideal month to visit Crater Lake! The two campgrounds within the national park are only open during the summer, but there are several wonderful campgrounds located on Highway 62 between Prospect and the park, plus Diamond Lake, located just northwest.
I suggest Natural Bridge Campground or one of the others nearby along the Rogue River.
No matter where you stay, make sure to look at the operating hours and road closures before your trip, just in case something has changed. You can find that information here, on the National Park website. If you decide not to camp, you can always park your RV at the rim and stay a night in the historic Crater Lake Lodge. We love it, and you might be able to get a reservation now that summer is behind us! Here’s more info.