You-- He first encountered Khan in 2267, when the Enterprise encountered the Botany Bay adrift in space. Kirk Prime had no idea what he'd just found or who he was dealing with when Khan was awakened from cryogenic stasis. At least, not at first. Information from that time period is fragmented at best, with many of the more official accounts destroyed prior to 1996 or lost in the Third World War. It wasn't until after Khan was already on board the Enterprise were the pieces of the puzzle fully realized and they discovered they had one of that period's most infamous dictators aboard their ship. Needless to say, Khan and the other Botany Bay augments very nearly commandeered the Enterprise, and would have succeeded in killing the other Kirk, if not for a moment of compassion paid towards him by one of his crew who had defected to Khan's side.

Instead of handing Khan and his people over to Federation authorities, Kirk decides not to press charges and instead leaves them to settle on Ceti Alpha V, quietly, where Earth and rest of the galaxy was out of their reach. I can't say what made him make this decision, but I imagine it was one he grew to deeply regret in fifteen years time.