If an alien is not using some form of respirator or breathing equipment, it may be assumed that they are deriving whatever they need from Earth's atmosphere. It may be oxygen, or carbon dioxide, or some other combination of gases and elements. It's possible that said elements are not available in sufficient amounts to keep the being healthy, and some form of supplement is being used off-camera. Higher concentration of oxygen can have a intoxicating effect on Earthlings - it's possible a similar effect may occur to alien races not used to the combination of gases in Earth air, tho I can't think of any examples at the moment. The Martian atmosphere is thinner then Earth's - it's possible that the richer atmosphere of Earth may contribute to a Martian's enhanced strength.
In the case of Daxamites, the amount of lead in the atmosphere is highly toxic, and builds up in their system quickly.
Tellus, from the Legion of Super Heroes uses breathing equipment, as he breathes methane. Quislet, as another example, is an energy being, and presumably requires no air at all.