When did airlines start using seat belts for passengers?
The world's first airline used Zeppelins, and there was no need for seatbelts.

In WW I seatbelts were standard issue for pilots, at least in Germany.

After the war, the airlines used seatbelts irregularly. Below are two pictures of the cabin of the Junkers F-13, the first all-metal passenger aircraft, which was first operated in 1919.



Junkers F-13 cabin, front view (picture source)



Junkers F-13 cabin, rear view (picture source)

While the belts on the front seats are clearly visible, the rear seats seem not to have them. If the picture date (Oct. 10, 1919) is to be trusted, they were used right from the start in 1919. The way they are fixed to the wicker seats, however, looks rather flimsy and will not give much protection in a crash.