A closed glass jar contains water vapor. The jar cools and liquid water begins to form. What will happen to the mass of the container as the liquid water is forming?

It will stay the same.

The mass isn't going anywhere (You did mention the jar was hypothetically and therefore hermetically sealed)

The only thing changing here is the density of the water in the jar (whether it be vapor, liquid or otherwise)

as the temperature drops and the vapour becomes liquid, the water will be more dense.

that's not saying that the weight won't increase as that is controlled by the force gravity exerts on the mass. (Hint: water is much more affected by gravity than air is)