If the ship that has Kylo Ren Crew on him dies but the Condition Card was already assigned before he dies does the condition stay in effect until the opponent suffers the crit?

Kylo Ren's condition card doesn't say anything about taking it out of play if the upgrade that inflicted it leaves play.

Rex's condition, Suppressive Fire, does tell you to do something with the condition when Rex is Destroyed.

Once a condition is assigned it no longer has any dependency on the upgrade that inflicted it (unless the condition itself refers to that upgrade).