I seem to spend a lot of time on the back of the tandem, and have had hand issues at times. These days my favourite mits have no padding, and I try and change hand position regularly. Making sure that the gloves are well fitting but not restrictive is important too.
Things I do to help include: Sitting up with hand off bars and stretching the wrists and arms sometimes (Thing 1 is a very tolerant captain and mostly puts up with the wiggly stoker). Paying attention to my hand position to ensure my wrists are not bent back - having stoker bars with flat tops (like the profile designs T2 wing bars) can help with that and stops you gripping the bars to tightly when on the tops.
When playing with my cycle computer and manipulating bottles I make a conscious effort to use both hands equally (after our second 1200km I couldn't do anything with my left hand for a week as I had been doing everything with my right hand).
Saddle position can also help, but I mostly find that my bottom complains before my hands if it is wrong.