Any kind of work at height is associated with danger and the need to hold the tool close to your hands, so as not to waste time on the descent and ascent. To work on the stairs, you can make a special carrying holder for the desired tool.
The device will be based on a steel plate, to which hinges of thick steel wire are welded. In the plate itself, you can also make a couple of holes, and at the end weld a long loop for the buckets so that they do not swing.
It is convenient to use such carrying on stairs with openings in the crossbars to place the foot of the device in them. The higher you work, the higher you can raise the necessary set of tools.