I had a few students who just couldn't help breathing through their nose. It just didn't help if I kept telling them to breath through the mouth so I had to try something else:
I tell them to stop playing completely when they need to breath. Then they take a deep breath through the mouth and continue. They will take the habit of breathing through the mouth after a week or two and won't need to think about it anymore. Of course you can't play like this in rehearsals or concerts but you can do it when you play by yourself. Using that method, the few students I had with this problem learned to breath through their mouth in about two weeks without me having to remind them every minute!
It is important to have a relaxed and fluid air flow while you play. I had the chance to have a few lessons with Mr. Vincent Chicowitz (former principal trumpet of the Chicago symphony) who taught me to practice just the air movement without any instrument or mouthpiece. Basically, you just practice your music blowing the rhythm as you would blow in your instrument if you were playing normally. Groups of slurred notes are blown like one long note like a tie over. You practice a difficult passage like this two or three times and then you go back to your instrument. It feels great! It really helps for passages in the high register. I personally do it every time I feel my air is a bit stiff.
I personally practice on my mouthpiece alone every day for a few minutes as part of my warm up routine. I do one or two octaves glissandos to increase flexibility and fluidity. It is also very useful to play a difficult passage on the mouthpiece and then play it on the horn. It allows you to focus on the actual movement of your lips and air flow. It's also a good way to find out if you hear the music correctly.
I am associate principal horn of the Singapore Symphony Orchestra and the author of the progressive methods. I'm happy to share my experience as a horn player and teacher with you.