You definitely do not need full on mountaineering crampons. In fact, their longer spikes often pose a tripping hazard, especially if you are not used to them.
Keep an eye on the weather. Winter precip can be extremely variable in the Southwest - every thing from huge snow drifts to bare ground. One or two trekking poles will also be very handy, even if the snow cover is minor.
The mules do a pretty good job keeping the Bright Angel trail fairly clear, as seen in the preceding pics.