I have to date never had an issue with access but it is designated as national park (mostly this is to do with river access)I am sure once more climbing happens here or bolting permission is sort the officials will hit you up for an access fee.
Just below Victoria Falls, the gorge forms an abyss between the countries with edges that drop away from the cliffs of both Zambia and Zimbabwe. Each successive basalt gorge is numbered in sequence starting from the youngest (First Gorge to the Fifth Gorge), followed by Songwe Gorge and finally the official Batoka Gorge; it is common for all these gorges to be referred to collectively as The Gorge or Batoka Gorge. Batoka Gorge is more than 120 km (75 mi) long with vertical walls that are an average of 400 feet high (the Zambezi river water levels fluctuates up to 65 feet between the wet and dry season). Batoka lies mostly within the Hwange Communal Land and is covered with mopane and riparian forests that are interspersed with grassland. On the Zambian side, the gorge is surrounded by the Mosi-oa-Tunya National Park, which contains a tropical rain forest that thrives on the eternal rainfall from the Falls. Victoria Falls National Park in Zimbabwe surrounds the other side of the gorge.