Unfortunately, there are no campsites near Dutch John within walking distance of the Green River. But there are plenty of nearby campgrounds in the area to choose from.
It will be hot in July, that's why I prefer the camp sites up on the rim near Red Canyon. They're much cooler in the summer, and the drive down to the river is not really that bad and there is plenty of parking at the river for large vehicles
If you're feeling really adventurous you could drive down to Browns Park and Jarvie Ranch, there's camping by the river there. The dirt road is steep, but usually well graded. The fishing is not quite as good, but there are fewer angler to contend with.
The Provo River around Heber City in northern Utah offers some great fly fishing and there's an RV park called "River's Edge" that is within walking distance. You might want to check that out. The surrounding Heber valley is very beautiful too, there are 3 state parks there.