Near the end of a long day of radio and TV appearances from home, Robert Kelly, an expert on East Asian affairs, prepared for another live video interview at 7 p.m.

As he finished up a soda and tightened his necktie before a Skype call from the BBC, he forgot to lock the door of his office in his high-rise apartment in Busan, South Korea’s second-biggest city, something he usually does.

That...