I think it's really something local to big US cities (or even smaller ones).
A friend who lived in Athens, Greece once told me how his sisters, who live in New York, came to visit him. He said that they were afraid to go on the tube, because they thought it wouldn't be safe. He said that they were used in the NY tube that they didn't think was safe.
I've had a few similar experiences myself (e.g. people being afraid of side streets in my cheery seaside town, where the biggest danger is someone from a hen or stag do getting sick on your new shoes). That's even more striking considering I don't actually meet that many US citizens. The few I do seem to always be looking over their shoulder.
I guess, the flip side of it is that people in Europe don't realise how safe our little region of the world is. We kind of take it for granted that just walking down a street in London or Paris, or other big European capitals doesn't automatically put your life and property in danger. For people living in places with high rates of violent crime, like the US, South Africa, Brazil or Mexico and so on, it's all very different.
I get that, in that sort of context, public transport is seen as unsafe- because anything "public" is seen as unsafe, and probably for good reasons. But that's not because public transport is inherently unsafe; it's public spaces that are unsafe in specific parts of the world. It's hard to decouple the two in peoples' minds though :/