Wells are covered to keep the water clean, not to keep it dry.
A well bucket rope is only going to be as long as needed, one end is going to get wet several times a day. Even in single person home well, you are probably collecting water for the home morning and evening, as well as watering the animals during the day.
Depending on how often water is taken from the well, it is most probably that one end is going to stay wetter then any reasonable amount of rain is going to make it.
If the rain is so significant that your rope is never dry, you won't dig a well you will catch the water from the roof of your house. Digging a well is a lot of hard work, building a gutter and cistern is much easier.
P.S. it does not matter if the whole rope or just one end is always wet. A well is going to have a rope just long enough to reach the water. If one section if it is wearing out, you need a new rope of just the right length.