How important is breathing? Well, no one needs to be reminded, really. As you read this your diaphragm is calmly expanding and contracting. It indicates that there is order in your life at this moment (hopefully).
I find it interesting how breathing can be quite telling as to one's emotional condition. We naturally associate long and slow breaths with relaxation, and heavy, fast breathing with a state of agitation. Thinking back over your life, I'm sure you can recall certain moments or activities where you breathed faster than what was comfortable. You had either just sprinted faster than ever before and couldn't catch your breath, or perhaps you had endured an emotionally traumatic experience.
For musicians, one's breathing must be relatively relaxed and controlled. It's impossible to be a good performer without composure, and that pertains to breathing, too. Obviously, it's most important for wind players, brass players, and singers. For us string players, it might seem immaterial because we work only with our hands, but it's just as important.
In ensemble playing, breathing together is vital. Tutti entrances and phrasing hinge on our sense of timing, which is largely dependent on our common breathing. We can play together because we all, as living beings, breathe in a like manner and have done so every second of our lives. A good conductor, of course, knows how to tap into it. A preparatory beat has to have a corresponding and appropriate inhale - otherwise, we won't know where the beat will fall.
You'll hear brass players breathe in unison prior to making an entrance. (Reminds me of serving on trombone audition committees and wondering why a few of the candidates used the instrument as a breathing apparatus - quite a loud sucking of air.)
Lately, I've listened to quite a few young bass players who have problems with how they breathe. I don't think it's uncommon, really, as many of us professionals grappled with the issue in our developmental years.
One issue I have confronted in students is this: the making of a constant and audible breathing sound while playing, by narrowing the air flow passage in the mouth, i.e., pursing the lips and raising the tongue. (It's like a half-whistle.) I think it gives them the feeling that they are intensely "into" the music.
Really, it's no more useful than breathing intensely while frying an egg. (Where you'd also want to keep relaxed for a successful flip…) What one is actually doing is tightening the diaphragm which can cause tension from the core muscles all the way down to the bow hand. This results in faulty tone production.
Other signs of tension are loud and fast breathing through the nose (sniffing), a clicking in the throat, and excessive humming along. There are probably whole tomes written about how to overcome faulty breathing for musicians, and I encourage further exploration on the subject. I haven't done any research, but I can make some suggestions based on my experience...
Hopefully, it's as simple as bringing it to the student's attention. If not, then experiment with ways of isolating it. Try having them regulate breathing while they play. It might help to have them count four beats per breath, or to fifty. I had fun recently when I asked a player to recite The Pledge of Allegiance while playing. It gave a different assignment to all of those muscles which were conditioned to tense up. It worked, to a certain degree, and we got a good laugh out of it. Improvised talking while playing might help, too.
There's also a piece for double bass called Failing - A Very Difficult Piece for Solo Double Bass by Tom Johnson which requires the player to speak throughout. This would be an ideal adventure for those with breathing issues. Also, one could run with that idea a bit and choreograph breathing for a piece or an etude - fast and slow breaths in different patterns or rhythms. One is then taking command and, hopefully, breaking the spell.
Over the course of days or weeks, exercises like these can help develop an awareness of breathing habits. Such "brain changes" usually don't happen overnight.
Finally, intense breathing can be associated with the emotionality of music. Encourage the student to break free of that. Try having them interpret a piece with a variety of emotional perspectives - ranging from boredom to enthusiasm. Sometimes, one needs to just lighten up!
If all else fails, get out the tape!