In general, I’m not that concerned about keeping fingers close to instrument keys. A lot of woodwind players and teachers seem to believe that “close” fingers mean more speed, which I haven’t found necessarily to be the case. To me, a much larger factor is tension: if my fingers are tense (because, for example, I’m trying really hard to keep them close to the keys), they move more slowly.
But when I work with beginning clarinetists (whether first-time instrumentalists or doublers), many of them seem to have a great deal of trouble with squeaks and with notes responding sluggishly—problems that I think in most cases can be traced to fingers not completely covering toneholes, or not covering them in a synchronized way. And one of the major reasons that this happens is that the fingers are too detached from from the keys. It’s not a question of distance, per se, but one of awareness.
One reason this is such an issue for clarinetists in particular is that so many fingers have multiple jobs. The left thumb operates a tonehole/ring and a key, which must sometimes be pressed individually and sometimes together. The left index finger has a tonehole/ring and two keys. The right index finger has a tonehole/ring and four “side” keys. And the pinky fingers have responsibilities exceeded only by the bassoonist’s thumbs. Throw in a couple of sliver keys, and you’ve got a lot of fingers constantly in transit from one key to another.