most of the time they aren't useful. They take more in maint. than they recoup in cities further out, and in close in cities they make a trivial difference in production. You have to be smart about what you are trying to build... just for an example, a tank might be 80 shields to build. If your city is turning out 45 shields and losing 5 to corruption, it doesn't matter b/c the tank will be built in two turns with or without a courthouse. Now, if the city is only producing 38 shields and losing 5 to corruption, a courthouse won't help either because its the total shields that are your problem... the tank will take 3 turns to build regardless. If you can't wait that long for tanks, use that city to build something like 60 shield artillery in 2 turns. And never build a courthouse in your capital/forbidden palace cities, thats a total waste.
Police stations fight war weariness, but you're almost better off just going with a dif gov't if you're planning lots of wars. Or try and build univrsl suffrg.