Yes, fleas breathe air and need oxygen to survive just like all other animals.
Fleas, like other insects, however, do not have lungs and instead breathe air through spiracles, or small holes and tubes that deliver oxygen directly to the cells using a process called diffusion.
Because they need air, fleas can drown or be suffocated. A great way to get rid of fleas off of your body for the body of an animal is to submerge it under water, such as by taking a bath.
It is an even better method if you can add a soap or detergent which will increase the likelihood of their death. The detergent decreases the surface tension on the water which prevents them from using it as a launching point with their strong legs, and jumping to safety.