How and/or when do beliefs influence the acquisition of virtuous behaviors?

You have raised very interesting, important questions, perhaps all the more overlooked for their simplicity. Here are some thoughts:

Aristotle, in his most famous work on virtue, the Nicomachean Ethics, argues that one becomes virtuous through diligent practice. You ask about belief itself aiding the acquisition of virtue.

As I’ve often pointed out, we have to distinguish two Sanskrit words which might each be translated as ‘belief’, astikyam and shraddha, both of which Krishna mentions in the Gita.
a) astikyam, from asti (he/she/it exists), is the simple faith or belief that God exists and is listed in chapter 18 as a component of brahminical action (brahma-karma sva-bhava-jam).
b) shrad-dha, literally ‘placing trust’ is discussed at 17.2-3. In my view, these are some of the most fascinating verses in the Gita. There Krishna explains that every person trusts something, and that in the dynamic act of placing (dha) our trust (shrad), we recreate ourselves.

It is this more powerful act of self-creation through deep trust that is more likely to satisfy the notion of belief influencing behavior.

In the  CC 2.22.62, Krishnadas Kaviraj formally defines the word sraddha, saying that “the word sraddha means trust, a very firm conviction [that] by rendering devotional to Krishna, all duties are accomplished.”

Bhaktivinoda Thakura also speaks of different levels of faith: soft, medium and strong. Clearly, the stronger the belief, or faith, the more it influences behavior. I would also mention that the belief-virtue dynamic is mutually interactive; it moves in both directions. By acting for Krishna, or even by acting in sattva-guna, and experiencing the positive results, one is brought to belief as much as belief may bring to virtuous or spiritual conduct.

In this regard, Krishna states in various places in the Gita that action in goodness (ie. virtuous behavior) leads to happiness, wisdom, etc. Thus, action creates belief, just as belief creates action. And, in all cases, there are degrees.

Also, I just thought of the important Gita verses 7.20-23 where Krishna states that He personally reinforces whatever belief (sraddha) we pursue, and makes that belief ‘unmoving’ (acala). Krishna does so not only internally by fixing a person on their chosen path, “whoever undertakes to worship whatever…”, but also externally by awarding a desired boon. Krishna states that He does this for those who actually practice a serious path.

So here is a case of “be careful what you worship.”

And how can people train their minds so that their conscious beliefs and thoughts have a greater influence on their behavior?

HDG: I suppose one must have integrity, in the literal sense: one must integrate (combine two things so that they become a whole) one’s belief and actions. Of course there will inevitably be a creative, dialectic tension between one’s present state, and the state to which one aspires. But the simple, and profound, acceptance of the need for integrity, will lead one on to wholeness, consistency, and integrity. One must of course also be very careful to cultivate beliefs which are reasonable and which reflect the purest, highest notions of the soul.