Level of Focus in Anapanasati

I've been practising meditation with the breath as an object. While meditating, I am mindful of whatever arises. I feel that such awareness gives insight into the mind. However, I am wondering whether one should investigate the mind this way, or try and increase focus by narrowing onto the breath. **Would narrowing onto the breath provide more benefit?** Hence, I wonder **how close to the breath one should be in meditation? Should *all* thinking be ideally diminished?**