Some studies have associated long-term consumption of chlorinated water with bladder and rectal cancer. Others suggest that it might also increase risk of coronary heart disease. I think it is not healthy to breathe the chlorine fumes that accumulate in and around the swimming pool. Mind too, that twice as much of a volatile chemical can be absorbed through your body's largest organ, your skin, than by drinking it. Swimming in chlorinated water can irritate your eyes, skin and hair, as well as the membranes of your nose, mouth and upper respiratory passages.
Chlorine gas is heavy, so it settles down on the water's surface.
When you're in a heavily chlorinated pool, you're breathing in more chlorine whenever you take a breath than someone standing off to the side of the pool. For that reason, personally, I don't swim in most public pools and places that smell very strongly of chlorine. If you do you might consider wearing a mask and snorkel. This protects your eyes from the irritant effects of chlorine in the water, and the snorkel will be somewhat higher than the level where the gas is densest. After your swim, leave the pool area and breathe fresh air to flush the gas out of your system.