From Neil Cohen's article "How to Score a Game," from John Thorn and Pete Palmer's Total Baseball, we discover that Henry Chadwick, one of the first newspaper journalists to take a literary interest in baseball, built upon a scoring technique devised by fellow New York journalist M. J. Kelly. "Chadwick created a minutely detailed scorecard so he would have a point of reference and recollection when he wrote his articles about the game," Cohen writes. He adds that Chadwick invented the modern boxscore.
Chadwick also invented the system we use to indicate fielders (pitcher=1, shortstop=6, right field=9, etc.), and the abbreviations we use for events (HR, HBP, BB, so on). Chadwick needed S for sacrifice, so he chose K for strikeout - K being the last letter of "struck," which was then in more common use than the term "strikeout."
Some people carry it further, using a K for a swinging strikeout, and a backward K for being caught looking. Some folks go with the more intuitive "SO," but this creates confusion with the abbreviation for "shutout," so "K" has remained the abbreviation of choice.