Do you need skirt boards on the stairs?
You don't have to install skirt boards.  Just like you don't have to install any molding in your house.  

However there is a difference here compared to your normal baseboards.


you are usually moving with more force on stairs.  
you are closer to the walls when on stairs.  You don't usually walk next to walls in your house much except for hallways.
if you damaged the bottom area of a wall by kicking it, well that is a pretty easy fix.  The drywall next to the stairs will not be as easy to patch and you may notice it a lot more.
you are often carrying things up and down the steps.  These nick this area and also cause your feet to hit the walls.
and then if you have baseboards already they look much better than when they meet a stair skirt.


Note that this is highly dependent on what function the stairs play and the overall finishes in the house.  If you are doing a high end house it is ludicrous to skip stair skirts.  On the other end if you have stairs going to an unfinished basement, generally not needed.