How would you reword this? Drop the her? You could be called sexist—or you could face a lawsuit from the woman who rode in that car who didn't think the warning applied to her. What about replacing the "his or her" with "their"? That may be less wordy, but unfortunately the grammar gods will look unfavorably upon you for disobeying their agreement rules and thus punish you by placing stray commas and apostrophes into your next report at work—which you handed out to 150 scrutinizing coworkers during a big presentation.

So what's a law-abiding grammarian to do?

Reword the sentence to avoid the awkward construction.

You could say, "Passengers should keep their seatbelts fastened at all times to protect themselves in the event of an accident."

My preference would be to simplify things even more by addressing the reader directly: Keep your seatbelt fastened to protect yourself in the event of an accident."

It has the exact same meaning, said in fewer words.