Because matter - solid, liquid, gas or plasma - comprises anything that takes up space and has mass, an experimenter can prove that air has mass and takes up space by using a balloon. According to About.com, balloons are inflatable and hold the gaseous matter.
The balloon increases in size, meaning that something is filling it. If air is capable of filling the balloon, it stands to reason that it must be taking up space. Now, take that same balloon and allow it to float in a glass of water. The balloon floats because the air inside it is lighter than the water. Next, draw a line on the cup to mark the level of the water and push on the balloon a bit. The water rises in the cup as the air inside the balloon takes up some of the space.
One can also prove that air has mass. First, suspend a meter stick from a string to create a balance and hang inflated balloons from each end of the meter stick. Next, use two pieces of tape to form an X on one of the balloons. Finally, use a needle to carefully make a hole through the center of the X so that air escapes slowly from the balloon. As the air escapes, one can observe that the end of the stick supporting the still inflated balloon will sink lower, which is proof that the air in the balloon has mass.