There are tons of options for you in this price range. Just Google Downtown Cincinnati apartments and you'll find many websites such as, http://www.forrent.com, http://www.rent.com, http://www.mynewplace.com, http://www.apartments.com I could go on for ages with all the sites.
With in each site you can put in your own criteria such as price, with W/D hook up's or even wash dryer included. I think you'll find some places that don't have hook ups might be worth looking at too! Check out 4th & Plum Apartments if you want a loft type living, they don't have hook ups in the apartments but their laundry facilities are really good and the additional amenities make these apartments a steal. I am not a big fan of Lytle Tower or Sycamore Place but others have found these to be great places to live. You may also want to take a look at Current @ The Banks, this is a new development still being built but phase one is supposed to be finishing up in the next month or two. There are plenty of options in and around downtown, don't rule anything out till you've checked it out because everyone's ideas and opinions will differ and what matters is what you think!
One Lytle Place is another good option: older property but it has been renovated to some extent. But I don't know about w/d in units. It's on the river tho' not all apts. face the river.
Good luck! If I can help further: Sheila@SheilaBell.com
I have this unit available in downtown Cincinnati.
1351 Broadway St $1250/mo 3bed,2.5 bath, avail now
Please let me know if you or your friend would like to view it. I own the property so there is no rental fee.