This paper considers secure communication over free-space optical (FSO) links suffering from turbulence-induced fading. In particular, we study the average secrecy capacity of applying physical layer security to achieve security in an FSO communication system with binary-level on-off keying intensity modulation and threshold detection, a widely used system in FSO communication. We first define the instantaneous secrecy capacity for our system and then provide a lower bound of it, which is achieved based on a simple design principle for practical applications and yields a simpler equation. We next derive a general expression of the average secrecy capacity and the corresponding lower bound based on bit error probabilities. Finally, numerical analysis is conducted for the special case of the correlated log-normal fading model to show the impacts of fading and channel correlation on the average secrecy capacity.