In waking life we use stairs and staircases to climb up or down to different levels. Most people, when they think of stairs, see themselves at the bottom looking up, so in dreams stairs are generally considered something to ascend, to reach up higher into our consciousness, traversing the levels of our mind.
With the above in mind, climbing stairs in your dream can show that you are reaching your life goals, and/or reaching a better understanding of your inner self and spirituality.
However, if you have a dream of falling down stairs it can represent failure or a setback in your waking life.
Walking purposefully downstairs is a common theme psychiatrist and therapists use in hypnotherapy to take you deeper into your subconscious, and dreams are no different.
If you dream you are at home, the place that is home in your dreams, and you find yourself running upstairs to your room or some other refuge, it could be that you are trying to escape the mundane aspects of your life and pursue the more spiritual, higher thinking side of your life. If you are running down these same stairs the opposite may be true?
If you are running up the stairs in fear, it could be that you lack confidence in your manual or everyday skills, in interacting with other people or a real life situation?
If you are running down the stairs in fear it could be that you lack confidence in your intellect, in your powers of higher thinking and wish for a simpler more practical existence?