Assuming you’re dealing with a species of dryland cactus,

not an epiphytic cactus,

follow these steps:

1. Remove all rotting roots and, if the damage has spread into the cactus stem, as much of that as necessary to have only healthy tissue left.
2. Allow the cut portions to develop a dry callous - similar to a scab - by setting it in a dry spot with good air circulation. A windowsill is a good spot as long as it doesn’t bake all day in the hot sun. This could take a couple of weeks or more.
3. Insert the cactus shallowly into an unglazed clay pot filled with free-draining potting mix. One that works well is made by mixing equal parts of 1) perlite (or pumice), 2) clean, coarse sand, and 3) standard potting mix.
4. Water thoroughly until water comes out the bottom drain hole. Never let the pot sit in water that drains into the saucer. Do not water again until the potting mix is completely dry.

The vast majority of cacti are adapted to environments that are usually dry. Knowing this should guide your care of them. If your cactus has rotting roots, it’s probably because the soil was moist for too long. This condition allows the roots to be attacked by predatory fungi and bacteria which, unlike cacti, do not survive well where the soil is usually dry.