Let me start with an example:
It's the waiting that was the worst. The attack would come, they just didn't know when. Could be another minute. Could be another hour. Ben sat nearby, sharpening his bayonet. Aaron wanted to scream at him to stop making that godawful noise, like a nail scratching glass over and over.
The fear in that short passage has a particular flavour: it's nervousness before an upcoming battle.
Now, every time your character is afraid, their fear would have a slightly different flavour, right? They're in a different situation, they're thinking different thoughts, they're afraid of a different thing. Even if it's, for example, a war novel, and your character is afraid multiple times in similar circumstances, he won't be thinking the same thoughts, would he? The fear of a green rookie is different from that of someone who has seen a few things.
Get into your characters' head. Figure out what exactly they're afraid of in each particular instance, what it is they're thinking. One isn't just "afraid to die". One might wonder whether it would hurt, or be concerned about leaving loved ones behind, or not completing some mission, or one might be more afraid of losing a limb than of actually dying... Or one might not be thinking of what they're afraid of, but of other things - past, future, hopes, failures, family... If the danger is not fully understood, one is very likely to be trying to figure it out. Etc.
The less time your character has to be afraid in each situation, the less time for introspection they have, and so their reaction could be more of one startled - a physical reaction. (A physical habit is also possible in other fear situations - @ErdrikIronrose gives a fine example of a habit that can be shown to illustrate an emotion.)
You can, and should, also rely on setting the atmosphere to convey danger. If the reader is aware of the danger to the character, you needn't say as much about the character being afraid. For example, if enemy planes are flying overhead, and the character is hiding, and maybe someone comments that those bombers are flying low, you understand that the characters in the situation are afraid.