It depends on the sickness. (And I haven't had to really address this yet. All of my children are very young.) So, I will share what I would do. We would have a really layed-back day. That might be the day we do science in the kitchen while making chicken soup and read a good book out loud.
It depends on the sickness. Generally, because they usually only get mild colds, we still school, or do partial days. For the more severe illness, we school the well ones, and I split time tending the ill child and schooling the other children. Only twice in the past years have I called a sick day for everyone (no school) due to everyone being sick (stomach flu and strep throat).
My children are a bit older (first, third and fifth grades) so when they are sick, they don't usually require my constant attention. Also, because they are used to sharing my attention during lessons, it feels natural to the children who are not sick to work without me or wait while I'm helping the sick child. We also have no time pressure (being as we miss any activities when someone is sick) so we take our time and have breaks whenever necessary. We may leave a few things for the next day if the child isn't up to it, but most of us complete most of our work. Usually when my children are sick, they like me to sit with and read to them, so we do this with school materials. We've yet to lose a day to sickness in three years. Even if we did, it would be easy to make up over a weekend. If I am sick, I am usually able to handle the few hours of lessons - it's not too taxing to sit and read with a cup of hot tea. But it's so nice to know there is always tomorrow and our schedule is so flexible if I need a day off.
We do best with routine, so even when we are sick, we still do preschool. However, I generally plan for easier, more fun activities. We do puzzles, board games, and fun-themed units. I also keep a stash of ready-made lessons so that when I am sick, I don't have to plan for the next day. Additionally, we shorten our school day a little as necessary when any of us are sick.
Because none of my children are really old enough for "independent study", we often do alternate activities like reading a book or watching an educational show like the Leap Frog shows or some of the PBS shows like Super Why! or Word Girl.
Fortunately we don't get sick often with homeschooling. We have had a minor cold, and a stomach bug last year. The stomach bug came during the holidays, so it didn't affect schooling. The plan is to play it by ear, if we have a bad bug to deal with. That is the beauty of homeschooling-flexibility!