Is moist air lighter than dry air? The simple answer is yes, but knowing why gets complicated.
Our air mainly consists of nitrogen (N) and oxygen (O), but there is also hydrogen (H). The mass of a molecule is determined by adding up the mass of each atom in the molecule. Water's molecular weight is 18; whereas, nitrogen sums to 28, and oxygen sums to 32. Given this fact, a water molecule is less heavy compared to a nitrogen or an oxygen molecule. Nitrogen and oxygen are heavier than they appear on the periodic table because the atoms don't naturally exist alone; instead, they exist in pairs. On the periodic table, oxygen appears as O2 and nitrogen as N2.
Any given gaseous substance, at a consistent pressure, volume, and temperature has the same amount of molecules regardless of the type of gas. Thus adding water vapor molecules to the air, creating a particular volume of air moisture, will allow lighter water molecules to replace the heavier nitrogen and oxygen molecules.
Dry air, primarily composed of nitrogen and oxygen molecules, has a higher density than moist air. Therefore, any specific volume made by water molecules will replace heavier molecules with lighter molecules. Because of this, it is true that moist air is lighter than dry air.