Like many of the answers you have already recieved, I am going to tell you that it is different, in each part of cincinnati. To be completely honest, there is REALLY good areas, and REALLY bad areas. I know you already have already gotton lot of answers, but I may be able to give you some different insight.
I grew up in kentucky, and went to a private school growing up, and now, I live in “over the rhine”. I stay there with my cousin and our boyfriends. It is a shit hole. I mean, as a near-20 year old, I can say that it is good experiance for me, and excelllent motivation for me to want to create something better for myself…..but, I would never EVER suggest you taking a small child near the down town area. Although the “culture” is nice, unless you want your kid to grow up and be a “thug”, I wouldn’t do it.
Instead try almost anywhere directly outside cincinnati limits. Like others have said, Anderson Township is nice, so is Indian Hill, and Milford or even Hyde Park is okay.
Northern Kentucky is really nice too. There is a very neat little town called Taylor Mill, not any more than 10 minutes away from downtown cinci., so if you still want them to experiance culture, you should try staying close, but just don’t get too close. You just might loose your child to the inner city.

Oh yeah, this is just an idea…. go to google local maps (just go to google.com and put “maps” in search engine) Then look at cincinnati in Satellite Mode. You can take a look for yourself, to see what it’s like. (it zooms in pretty close….kinda creepy….makes you wonder who’s watching.)

Good luck in your moving adventure!