Learn English – Make something great out of a bad situation

The closest I can find to what you're describing is "news-induced" or "media-induced" anxiety. A quick search does not indicate widespread adoption of this as a catchphrase, but it seems like a sensible fit.

"Stranger danger", as you cite, is something that arose out of a particular kind of bad news that was prevalent at the time. I am not aware of any similar catchphrases around computer hackers, spying, terrorism, etc.

In mental terms, someone excessively worried about such things might be "hypervigilant" or suffering from "anxiety disorder". Or just plain "paranoid".

There are some generic idioms about "jumping at every shadow" or thinking that "danger lurks around every corner", but I think those are probably too broad for what you're looking for.

A lot of words or phrases move away from the literal meaning and this literal meaning is bleached or ignored but instead the focus is on the function to express a kind of attitude. Some people call this pragmaticalisation; the phrase serves a pragmatic function of expressing an attitude. Out 'good-bye' or bye started life as 'God be with you' - which originally invoked a blessing on the other person. And people typically invoked a blessing when they parted, so it became associated with leave-taking and the literal meaning was forgotten. We can also think about how 'thank you' often functions not to thank someone today.

'Bless you' has also undergone pragmaticalisation. I would say that 'bless you' expresses a degree of fondness for some people who have disappointed you. In my own usage, it is similar to how I might say of say of children, 'They are uncontrollable, bless their hearts.' I would imagine that it started life as an actual blessing (as mentioned by bobtato). You are typically fond of the people you want to bless, but the usage mentioned, the fond attitude is preserved and the actual blessing forgotten.