Why does Lord Harkon do nothing when I accept his offer of becoming a vampire?

This is a (sadly) common bug. Usually the only fix is to reload a previous save. But what works (PC only as far as i know seeing you need access to console commands) is to go to the entrance door, activate console commands (press ` next to the 1 on your keyboard) click on the door in the console command menu and then type unlock go outside and then back in and the quest semi resets.