I'm not sure whether and to what extent this information conflicts with simchona's answer, which focuses on "swinging strikes" (that is, attempts to strike the ball that fail to touch the ball or that cause the ball to go foul), but the so-called "called strike" (in which the batter doesn't swing at a pitch in the "strike zone" above home plate) appears to date back to the late 1860s at least.
From "Our National Game: The New Rules," in Our Boys and Girls, volumes 5–6 (1869) [combined snippets]:
Rule III. The Batting Department.
The batsman must stand astride the home base line when striking at the ball. If he does not, "foul strikes" must be called on him, and three such strikes put him out. No base can be run on a "foul strike." Batsmen must strike in rotation, and any one absent when his turn comes must be decided out, unless absent by sickness. The first to go to the bat, after the first inning, is the player next to the one who was third out on the previous inning. If the batsman refuses good balls, a strike must be called ; on third called strike he must try to get his base; but if the ball be caught on the first bound behind, he is out. The striker is out if a fair ball' be caught on the fly, or be held at first base before he touches the base ; if at the same time, he is not out; or if a foul ball be caught on the fly or bound.
In other words, the rules of the game of "base ball" in the 1860s provided that a batsman could strike the ball "fair" (by striking it while standing astride the home base line) or "foul" (by striking the ball while not properly astride the home base line), or he could have a strike called against him (by not swinging his bat at "good balls"). Conspicuously absent from this account is any mention of "swinging strikes"—attempting to strike a pitched ball but missing it altogether—perhaps because, in the early years of the game, missing the ball altogether was a relatively rare occurrence, since "the ball was pitched or lobbed to the plate underhand," according to Peverelly's National Game.
That description also helps explain why the verb pitch, rather than throw or hurl, was used to describe how the pitcher released the ball toward the batsman: In the early days of the game the pitcher's action was closer to tossing a horseshoe than to throwing a stone.
In the case of a "called strike," it isn't clear whether one strike is called for each "good ball" that the batsman refuses to offer at or whether it is a cumulative punishment for taking too many "good balls" in one at-bat. But it does seem clear that the word strike is applied in this situation because the failure to swing is treated as if the batsman had swung at the ball from the wrong position on the home base line and therefore been guilty of a "foul strike."
Edmund Routledge, Every Boy's Book: A Complete Encyclopædia of Sports and Amusements (1869) repeats (with slight variations and greater detail) the rules outlined in "Our National Game," and then discusses situations where the "striker" either doesn't swing at a pitch at all or swings at it but misses:
Should a striker stand at the bat without striking at fair balls, for the apparent purpose of delaying the game, or of giving advantage to a player, the umpire, after warning him, shall call one strike, and, if he persists in such action, two, and three strikes. When three strikes are called, and the ball be caught, either before touching the ground or upon the first bound, the striker shall be declared out, provided the balls struck at are not those on which balls [pitches that are not "fair" to strike at] or balks [illegally thrown pitches] have been called, or not those struck at for the purpose of willfully striking out. If three balls are struck at and missed, and the last one is not caught, either flying or upon the first bound, the striker must attempt to make his run, and he can be put out on the bases in the same manner as if he had struck a fair ball.
Evidently, the justification for using the verb strike to describe a swing in which the bat doesn't touch the ball, or to describe a failure to swing at a fair pitch at all, is that the actual or ascribed action isn't "striking" (making contact) but "striking at" (attempting to make contact).
