I personally wouldn't worry about it. But that's just me. I spend a lot of time up on the rim every year, and I've never personally encountered a bear, though I have certainly seen evidence of their living there (footprints, scat, markings).
Dane Canyon is a beautiful area, and there's ample water in it, despite the overall dryness of the forest right now. While you may encounter wildlife at any time when recreating outdoors in Arizona, big animals such as bear, mountain lion, and even elk and deer, typically do their very best to avoid people. Even if you were to encounter something, if you take a few precautions, (especially regarding food storage) you should be just fine.
Like I said, that's just my opinion. If you're unsure or uncomfortable about it yourself, do what feels right to you.