Everybody has different fears. Not everybody is afraid of death (or not everybody admits it). Not everybody is afraid of sharks. Not everybody is afraid of getting struck by lightning.
Not everybody is afraid of getting cancer from smoking, despite what anti-tobacco wants to do. In Australia, they've passed a law that no brand names will be on cigarettes, only the horrible pictures of diseased organs. In the United States, the court said "yeah, no, you won't be doing that diseased organ thing." Really, we've known for many years what cigarettes can do. Do they do it to everybody? No. But there's always the chance.
Not everybody is afraid of The Bomb. Frankly, not everybody even knows to anymore. I was alive during the Cold War, sure, but not really aware of it. I was alive when the Berlin Wall came down (and recently bought a piece of it at Salvation Army for $5!) but didn't know. I know now what Duck and Cover is; I know now what happened at Chernobyl (well, as much as any non-Soviet civilian might). I've watched nuclear test footage, I waited at the edge of my couch when the Fukushima reactor was going up. I'm nervous about Iran and North Korea having The Bomb; I don't understand how people can think that The Bomb Isn't a Big Deal. Japan certainly hasn't forgotten it. I've watched enough anime to know that.
Not everybody is afraid of monsters, per se, but many people know and are fascinated by man become monster (serial killers, spree killers, kidnappers, terrorists). Not everybody is afraid of "nature" as such, but there are certainly enough Natural Disaster movies that made a bajillion dollars (The Day After Tomorrow, The Perfect Storm) that it still works.
A story that makes people afraid, either of the situation itself or for what happens to the characters, is a good one. If you can evoke that sympathy from your audience, you've succeeded either in some small way, or in a major one. Good writing is visceral, and makes a certain mindset happen. It dazes the reader when they look away from the page, because they've been transported by your words.