The first use (in any media) was in the radio play "The Scarlet Widow" from 1943. The eponymous villainess steals kryptonite from the Metropolis Museum and uses it to build a cage for Superman, incapacitating him for several episodes.
Interestingly, the plot device was apparently intended to give the Superman voice actor (Bud Collyer) a few days holiday as his character was restricted to pre-recorded grunts and screams.
The first use in any comic serial was from Superman #61 ("Superman Returns to Krypton"). Superman fights a self-styled Swami whose kryptonite turban-stone causes Superman to become weakened. This edition is also famous for being the first time that we see the planet Krypton in a non-exploding state.
