You don't say where you will be coming from, but I'm guessing Portland or San Francisco, so Medford would be the easiest. I'd pick Medford over Klamath Falls, since the road from Medford to the south entrance of the park is the one that is open year round.
Medford probably has better eating choices as it is a somewhat larger city than Klamath Falls. Medford is also just 20 miles from Ashland where they have the famous Oregon Shakespeare Festival, if something like that would interest you. The plays are presented until late October.
The Hellgate Jet boat tours from Grants Pass are great fun. The weather should be warm there, you will get wet! We took the dinner tour--great food, if you leave hungry, it's your own fault!
The other Rogue River option is the jet boat tours from Gold Beach over the the Oregon coast. They go upriver to the little town of Agnes. It will probably be warm in Agnes, but when you start the tour in the morning it can be rather chilly. We've done both (on the same trip actually) and enjoyed each of them.
We just did the Gold Beach lunch trip with Jerry's and loved it. Lucas Lodge was wonderful for lunch. It is an additional $12/pp for lunch. :).
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