Despite the work of intrepid researchers, we'll probably never know when a batter first stepped to the plate wearing something to protect his skull from errant pitches. We do know that Roger Bresnahan -- a catcher famous for popularizing the use of shin guards while behind the plate -- was hospitalized in 1907 after being struck in the head by a pitch, and that when he returned to the lineup, he wore "a primitive helmet known as the Head Protector, invented in 1905 by Frank Mogridge and marketed by the A.J. Reach Company."
Bresnahan's experiment didn't last long, nor did others over the following decade or so. In 1920, Indians shortstop Ray Chapman died shortly after being hit in the head by a Carl Mays fastball. This did not, however, result in a baseball-wide movement toward protective headgear. Cleveland players did experiment with leather, football-style helmets in the spring of 1921, but "the players found them uncomfortable, and the effort was abandoned."
The next blip came in 1937. On May 25, Tigers player-manager Mickey Cochrane was beaned and nearly died. Perhaps as a result, Senators owner Clark Griffith ordered a supply of helmets designed by James "Foulproof" Taylor (nicknamed "Foulproof" because of a protective cup for boxers he'd invented). The helmets arrived on the last day of July. On the first day of August, Washington Post columnist Shirley Povich wrote, "It is now Griffith vs. his ball club, because the players, after taking a look at the lopsided caps, vowed they wouldn't wear the dizzy-looking things. They'd rather get hit in the head, they said."