- The pilots seem to be even more aggressive than usual about keeping the fasten seat belt sign on, even when there was clearly no turbulence.
- Passengers were blatantly ignoring the sign and getting up and walking around and the flight attendants weren't making any attempt to stop them.
I doubt these are unrelated.

Would this behavior reduce the airlines liability if a traveller were to be hurt during some sort of unexpected turbulence?
Perhaps more importantly, do the pilots have any incentive to reduce their airline's liability?
Do you mean that you doubt that they're related--in the sense that the passengers acted more or less as they would have done had the pilot followed the "normal" pattern? Or that the continually lit sign seemed to make the passengers even more antsy than they would have been otherwise?
By the way, I don't know if the pilot was necessarily being "aggressive" in his use of the seatbelt sign. As I understand it, pilots tend to turn the sign on in response to reports of turbulence in certain spots from other airplanes that have just previously flown the same route. The advantage of this anticipatory approach is that passengers are prepared for a sudden onset of turbulence. The disadvantage is that sometimes the reported turbulence dies down before the airplane arrives. Although such false alarms are obviously somewhat inconvenient, it seems unfair to blame the pilot, if this is indeed what happened. The pilot, after all, may well have had good reason to suspect, the whole time, based on the previous reports, that the turbulence would begin "any minute now".
My (non-empirically backed) impression is that pilots have started to use he fasten seatbelts sign more aggressively over the past few years (I'm tempted to say since 9/11). This probably produces an initial reduction in people walking around, but it seems to me that over the same time period people have started to feel like it's always on and so started to ignore it.
Fasten Seat-belt means nothing. The words you want to listen for are "Flight crew please be seated" which is often said very quickly.