If our main characters aren't brave, if they don't have the courage to confront evil, face their fears, and fight back to defend themselves or their loved ones, they're not likely to be sympathetic characters. Most readers of mystery, suspense, and thrillers expect the main character to live up to the challenge, to be a hero.
From Agatha Christie's Hercule Poirot and Miss Marple to Lee Child's Jack Reacher, bravery is one of the most important traits of the genre's protagonists.
Interesting enough, bravery in a main character is often accompanied by a reckless disregard for his own safety, and sometimes a big streak of stupidity.
In my two amateur sleuth mysteries, The Prairie Grass Murders and The Desert Hedge Murders, 60ish characters Sylvia Thorn and her brother Willie Grisseljon are actively involved in solving crimes. Sylvia, although she should know better, impulsively sticks her nose in places it doesn't belong. She has a heightened sense of responsibility and wants bad guys apprehended. If she thinks the law isn't up to it, she steps up to the plate.
Willie, a more cautious soul, would prefer to leave the sleuthing up to the police. He tries to talk Sylvia into abandoning her detecting activities, but when the chips are down, he'll do anything to keep his sister safe.
Have you ever read a book from this genre where the main character gave in to his fears and never showed the slightest sign of courage? Would such a character be appealing?
"Bravery never goes out of fashion."
--------------------William Makepeace Thackery (1811-1863)