We would like to find a nice hotel in the Latin Quarter for about $200 a night. Doesn't need to be fancy but safe, clean and with some character if possible.
Three hotels we've stayed at and liked in the Latin Quarter, and are in your budget, are: Hotel St. Jacques - we've stayed here twice, had room with a balcony which we really enjoyed but all rooms are nice Hotel des Grandes Ecoles - nice garden Hotel du Pantheon - in your budget depending on when
We stayed at the Hotel Home Latin in the fall of 2011 and would happily do it again. It is on a quiet side street convenient to Notre Dame, Luxembourg Gardens, the Panthoen etc.
I stayed at the Familia Hôtel in the heart of the Latin quarter back in 2008 and absolutely loved the experience. It was the little things I loved. If you want character, this is your hotel! http://www.hotel-paris-familia.com/
I second Alexander's recommendation of the Familia Hotel. The Familia is not on Rick's list, but neither are a lot of good hotels. It is north of the Sorbonne, in a quiet, more traditional part of the Latin Quarter, away from the tacky strip of tourist traps and heavy traffic along the other side of the quarter.
We loved Hotel Diana.