"Strike" comes from an Old German word meaning to "touch lightly", though by the 13th century it had acquired the sense of "hitting hard". The root was the Indo-European streig - or stroig - the source of the Latin strigilis, a tool for scraping the skin after bathing.
Organised sport was quick to take up the word, and its offshoot, "striker". In 1891, WG Grace wrote in his book Cricket that "It is the striker's duty to call [for a run] if the ball is hit in front of the wicket". The OED Supplement records an 1853 baseball reference in the Oregonian newspaper: "No doubt they will find that the strikers have struck out."
The OED Supplement's first record of "striker" in a footballing context is from an unexpected source, in the 1963 book Soccer, although it must have been used before that, when the five-man front line became obsolete. "If John White or another Spurs player is bringing the ball up... I move into a position ready to race through... It is the ball goal-strikers dream of." Step forward one of the game's greatest strikers, Jimmy Greaves.