There is a time for mouth breathing. Playing saxophone is easier with the faster air flow, not to mention in a heated verbal debate or song singing. Panting seems to be a natural reaction in athletics to increase oxygen. I've even dedicated an entire line of study to various forms of breathwork, some of which encourage the use of mouth breathing during the meditation for that very reason.
What if i told you, except for these rare exceptions (which in some cases can be overcome), you should always breathe through your nose?!? After years of research and personal independent study, I am about to make this case for you: Don't be a mouth-breather.
Watch the Breath
Articles on mindfulness meditation.