The question of when to visit Livingstone hinges almost entirely on what you want to see at the falls. The Zambezi runs high from February to April (the wet season), and the volume of water at its peak is extraordinary — but the spray is so dense that visibility is severely limited. You will get soaked, hear the roar, and see only glimpses of the actual falls through the mist. Dramatic, but not the panoramic view of the postcards.
May to August is the dry season and the peak tourist period. Water levels are lower, views are clear, and activities like white-water rafting and the Devil's Pool swim (from Livingstone Island) are possible. The weather is dry and warm by day, cool at night. September and October are the hottest months before the rains return, with temperatures above 40°C possible.
For the full picture — substantial water flow and some visibility — late May through July is the best compromise.