If Player A has a unit, and then Player B uses Lotus Trap on Player A's unit, Player A reacts by using Ki Barrier. Does Player A determine the order of the replacement effects?

Yes, Player A determines the order.

Both Lotus Trap ("double all damage") and Ki Barrier ("prevent the next 7 damage") are replacement effects that apply to the same event — damage being dealt to the unit. Per Rule 372, when multiple replacement effects apply to the same event, the owner of the object being acted on chooses the order they apply. Since the unit belongs to Player A, Player A decides.

This choice matters significantly. For example, if 5 damage would be dealt:

• Ki Barrier first, then Lotus Trap: Ki Barrier prevents all 5 → 0 damage, then Lotus Trap doubles 0 → still 0. The unit takes no damage.
• Lotus Trap first, then Ki Barrier: Lotus Trap doubles 5 → 10, then Ki Barrier prevents 7 → 3 damage gets through.

Player A will almost always want Ki Barrier to apply first so the prevention happens before the doubling inflates the damage. Either way, the order is Player A's call.