Walking — and other exercise — can benefit the brain. Walking improves blood flow to the brain, it protects blood vessels, it reduces the risk of things that could be bad for the brain, like hypertension, obesity and stroke — so at the end of the day, pretty much any form of aerobic physical activity would probably have similar effects. It's cheap and it's much better than picking drugs that have side-effects, and it probably has many good effects, whether it's related to your brain health or not.