Ed,
Shepherd's Crook Golf Course in Zion, IL is the rare public (municipal) course that provides carts, but actually promotes walking. Their website states that walking is not only permitted, but encouraged.
Now, I have no idea what effect (if any) this approach has on their ride/walk ratio. I've only played the course once, so I don't have a feel for their typical customer. Sven Nilsen has quite a few rounds up there, so maybe he can weigh in regarding what he has seen from the other golfers at SC.