Original sin is not a personal sin, so in this regard, a baby has no "sin". But if we define "sin" as something without God, then we can rightfully call original sin as such because we are born without sanctifying grace, God's presence in our souls. As such, this state, sans God, is called "original sin". It is a lack of God, not a personal sin.
Regards
Besides, "dying to sin" is a figure of speech that refers to all sin, past, present or future, not merely one that has already been personally committed. In baptism we filfill the death as precondition of new life: "unless the grain of wheat falling into the ground die itself remaineth alone. But if it die, it bringeth forth much fruit" (John 12:24-25). As a rite of initiation it is looking forward, not past.