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AFAIK it's only dangerous for mechanical watches. If the spring gets magnetized, its windings will stick together so it can't work properly.
I remember as a child my very first watch was mechanical. On the back it read "antimagnetic". I wanted to know if this was true an tried it with a strong magnet. The watch WAS attracted by the magnet - I was disappointed. Only minutes after, the watch stopped.
Luckily my grandpa was a hobby watchmaker (he had given me this watch as a gift only a few days before) and he had a demagnetizer. He showed me the magentized spring and explained what had happened. I don't remember though what that "antimagnetic" actually meant.