I'll be honest, this answer surprised me when I looked up the info about it.
Anatomy and Physiology
First off, it's useful to know that you actually breathe through one nostril at a time already, and this changes about once every 4 hours. This is called The Nasal Cycle, and basically put, your body blocks one nostril and clears the other, and swaps round regularly. There is no muscle to control which nostril you breathe through, so you can't independently decide to breathe through a selected nostril without manipulating the airway manually. This differs a bit from mouth vs nose breathing, as you can choose to breath only through your nose, even with your mouth open, using the Superior Pharyngeal Constrictor muscle. The major opposition to the claim that Yogis breathe through an selected muscle in order affect their state of mind is that you'd literally need to hold one closed, which doesn't seem too conducive with other activity that may be going on, however...
Literature on the subject
Despite my reservations, there is confirmatory literature on this subject. This study found that subjects tended to perform better at spatial tasks whilst in the left nostril phase of the nasal cycle, and better at verbal tasks during the right nostril phase, which aligns with the original claim you asked about.
As further back up to this, this study took 2 groups of individuals, gave them yoga training for one month, and had one group concentrate on right nostril breathing and one on alternate nostril breathing, a number of times a day across the experiment. Both groups showed an increase in oxygen usage, but the right nostril breathing group showed a gain of around 200ml/min over the training, where the both nostril group showed an increase of around 100ml/min. Statistically significant changes were also noted in the change in galvanic skin resistance, heart rate, and respiratory rate.
More recently, this study looked at physiological effects of left nostril breathing, right nostril breathing and any nostril breathing, and found that there were significant differences between each, finally concluding that there may be potential for use of this for therapeutic applications.
I've looked through a few more studies too, and all have concluded that a physiological difference is observable. Those that have observed mental performance have also noted differences in ability at tasks, and those that addressed mental acuity have all concluded either that right nostril breathing performs better, or that right nostril and any nostril breathing performs better than left nostril alone.
Although, having looked at the evidence, I'm satisfied that there is an observable difference in several physiological signs, as well as in mental performance, I'm still not clear on how Yogis manage to 'select' a nostril to breath through. The studies I have read haven't explained anything further than to say that Yoga breathing techniques were used which involved concentrating on a nostril, and I'm unaware of a physical mechanism within the pharynx and sinuses that would allow this to happen.
It's possible that Yoga techniques can allow a person to 'cue' their nasal cycle to change to a preferred nostril, but I've not seen a physiological study of the nose and sinuses during these activities, so I can't say whether that's the case. This said, it's therefore possible that simply concentrating on breathing through a particular nostril could make these differences. This would only be contradicted by the first study which found differences at different stages of the nasal cycle, and didn't attempt to manipulate which nostril a subject breathed through (they waited until the applicable nostril was in use, essentially).
Overall verdict - It's possible, and there is some literature that backs it up, but a little more knowledge of what is occuring in the noses of Yogis is needed!