The work done against gravity (= increase in potential energy) is the same in both cases. Extra work might be done to create kinetic energy.

If the walker and runner both start and end at rest, then overall no kinetic energy is created in either case. All of the kinetic energy which is created is used up to increase the potential energy.

If the walker is still walking and the runner is still running at the top of the stairs then overall some kinetic energy has been created. The runner has created more kinetic energy than the walker because he/she is moving faster at the finish.

$$\text{total work done by person = increase in potential energy + increase in kinetic energy}$$

The person who runs up the stairs finishes with a greater increase in kinetic energy, so the total work done by the person is greater. 

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**Note** that it is the **overall increase in kinetic energy** between start and finish which matters. It makes no difference how the kinetic energy has varied in between. In physics terms, person A who walks very slowly for most of the way then runs the last few steps does more work than person B who runs most of the way then walks the last few steps. Person B will probably be more tired than person A, but this is a separate issue about the difference between external physical work and internal biological work. See [Why does holding something up cost energy while no work is being done?](https://physics.stackexchange.com/q/1984)