There are a lot of apartments in the area immediately surrounding NKU's campus, and I would imagine that the prices are reasonable since most of them cater specifically to students. If I were you, I'd focus my search in Highlight Heights (where NKU is located), as well as adjacent towns like Cold Spring, Ft. Thomas, and Wilder. All these places would put you within 5-10 minutes of campus. If you want to get a little farther out, you might try Alexandria which just a little bit south (about 10-15 minutes from campus). Southgate also is pretty close and probably has the cheapest apartment options, but it's a bit seedy compared to the other towns I mentioned.
Edit: I did a search on the Cincinnati Enquirer's website for 1 bedroom apartments that allowed pets in Cambell county, and here's what came up:
http://www.apartments.com/partner/Re...nebdrm=1&am4=0