Horn handles only require a polish with brasso now and then to bring back the lustre. Shanks will have received three coats of polyurethane varnish and tipped with a brass ferrule before leaving my workshop and should only need a wipe down with a damp cloth, though if used heavily the shank can be re varnished.
The most important rule with any horn handle or walking stick is not to leave it in strong sunlight, especially do not leave it in a car on a hot sunny day. The reason being that heat has been used to form the horn handle, and while it should not move away from its final shape once it has been made, there is always the possibility that if heated sufficiently by a heat source such as the sun, the shape of the handle will alter and try and revert back to its original shape. The shank will also have been heated during the straightening process.