A balloon displaces the certain volume of the air, on the surface of the Earth. If the Sum of the Weight of the balloon and its parts is lighter than the volume of the air displaces by it, then the balloon will rise.
Now, If the Volume of the Sum of the Weight of the balloon and it parts is heavier than the volume of the air displaces by it, then the balloon will fall.
The acceleration due to gravity of the Moon is 1/6th that of the Earth, and also there is no atmosphere, thus, the weight of the balloon and its parts is obviously heavier than the volume it displaces in the Surface of the moon (Since, there is no atmosphere, thus, Zero volume is displaces by the Balloon.), therefore, the balloon will fall on the surface of the moon.
Also, there is no air in the moon, thus, there is no air resistance. Therefore, acceleration by which the balloon will fall on the Moon is 1/6 of 9.8 m/s²(Acceleration due to gravity of the Earth) = 1.63 m/s².