Stairs symbolize ascension. Ascension can be progress towards a more developed self, a higher state of awareness or consciousness, or entering into a new life. If you dream of someone climbing stairs, it could mean that the person is transitioning between this life and the next, such as what happens at death.
Going down stairs can symbolize an exploration of deeper, repressed thoughts or memories that may cause fear or discomfort. It may be time to explore unresolved experiences from the past that currently bother or haunt you. Although this process can be scary, if you are honest with yourself about your thoughts and emotions, it can ultimately lead you to greater peace.