This post collects a lot of miscellaneous observations and advice, some from other thinkers and some from myself. It's all basic-level.
There's plenty of stuff out there
far more advanced than this. This is not written with any particular game system in mind, and it includes a mix of game master advice and game designer advice.
Here's the fundamental problem of this topic: most of the time, preserving the players' agency is paramount. But fear complicates this priority. Fear is an involuntary mental state, but it can shape your behavior in profound ways. No heroic adventurer would choose to be afraid when faced with peril.
Ideally, you trust the players to roleplay their characters' emotions on their own. "If it seems like your character would be afraid of this, then try to play them like they're afraid." And if everyone is participating in good faith, they'll try their best. But unlike other emotions, authentically roleplaying fear is much easier said than done.
There are a number of ways to help resolve this problem. Different games and playstyles offer their own answers. Some of them contradictory, some of them mix well. Here's the stuff that makes sense to me based off of all my experience. I'm splitting this into three sections: 1) Player Fear, 2) Mechanical Fear, and 3) The Overlap.