Yes the work is done, when the porter is walking along a road or he is climbing up or down.
When he is climbing down also he has to work against falling down for which he has to generate an energy. He has to work against the acceleration due to gravity and manage himself to keep a uniform speed down against gravity every moment.Actually should maintain a zero acceleration against the acceleration due to gravity.The larger the steps the more energy is required as his centere of gravity moves up and down more and the PE difference is high in each step. Also the speed or the number of steps per second also adds to the kinetic energy. So his body works in the walking also.
In walking along a load on his head he appears to move uniformly . But his movement is not uniform but descrete. For every step he pushes the ground to get a reaction which helps to get an instant acceleration from zero to a small per second and rest and repeat the activity of getting this acceleration. During each step his centre of gravity along the load moves smoothly along an arc (whose radius is equal to the length of his leg) up and down. So a potential energy differemce occurs for each step. The more the number of steps, the more the energy consumption or work done.
If we look at practical reality, the body of a porter carrying load on head on a level road uses more energy than is required when the porter is resting completely. In this way the porter is doing some work.
However, when we think of theoretical definition of work, there is no force resisting the horizontal movement of the load when carried on a level road, and therefore there is no work done when a porter carrying a load on his head walk along a level road.
When the porter is going down the stairs the gravitational energy acting on the load is assisting the downward movement of the load. In this way energy is released by downward movement of the load. Thus the porter is doing negative work on the load.
There are no one clear answer to the question. For example we can complicate the issue further considering the kinetic energy that the load should have gained while coming down. But in reality the load may not have full amount of the kinetic energy as the rate of decent of the porter is likely to be less than the downward velocity required to give the load required kinetic energy. In this way we can say that the porter is doing some work in resisting the downward gravitational pull that would have resulted in the load having the amount of kinetic energy when falling freely under the influence of gravitational force.