Princess doesn't cater to kids, as does Carnval, Disney and RCL, but plenty of kids cruise with them. And as it's August, school is presumably still out, which means there will be even more kids.
We cruise with Celebrity, which has a similar demographic to Princess. My kids started cruising when they were 5 and 7 years old. We also did Carnival, RCL and NCL. They preferred Celebrity. Better food, better service. You will have as much fun as you allow yourself to have on Princess. It's also a good opportunity to eat things you have never tried. If you don't like it, you can get something else. My kids are foodies, so they really liked trying all sort of different things when we cruised.