I've just had a chat with one of the lads regarding digs in Athens, who has unknowingly booked in not the best of areas. Without taking over the mantle of Kerem, I have some honest advice having visited the city several times and having family there.
As Brits, we always gravitate to the center where the sights are, like many did in Istanbul and were left disappointed in areas south of the river near Fatih. I'd say try to avoid staying in Omonia or Exarcheia if you are after a central location. These areas are very dodgy for Brits who will stand out a mile as wealthy and easy targets for the local migrant population, lots of petty crime and fraud in these areas daily. Whilst cheap and convenient hotels are in abundance around these areas, it's not safe after dark. It's where you'll find the brothels for those that way inclined but go in groups and expect to be approached and hassled by non-Greeks. Much better locations would be Plaka or Kolonaki if you want a central location and still a stone's throw away from the sights at Acropolis and Monistiraki. but expect to pay a bit more.