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?
Yes, once you have reached sufficient level of concentration you may continue on the breath (samatha) or investigate the mind (vipassana). In my own practice I have a preference for reaching the calmest level possible, so I simply stay with breath, but more quietly, gently, less control, just allowing it to be, no more. At the same time, I don't ignore or suppress things as they arise, i just let them arise, fall, arise, fall. In one door, out the other. It's "kind of" a combination of both Calm and Insight.