Governors' traits can go up as well as down in cities but you have to micro-manage them. Personally, any new family member I get with the budding bureaucrat trait becomes a governor, the others field generals. The governors go to where trade is good, or I'll be doing a lot of building, & they tend to pick up positive traits which either save or make me ££. I think the key to avoiding negative is making sure that tax/public order ratio is optimal, religious unrest is under control, & essentially a good job is being done. I find it almosr as interesting to try & build these governors as my generals.