Firstly, make sure that you check all stairs on the property, not just the main staircase. This includes any stairs you have outside. Check exterior stairs to see if water pools on them, this may mean they do not have the right amount of slope.
Are the stairs up to code? Bannisters should be the right width apart to ensure that children cannot get their heads struck, and stairs need to be of equal height to stop them becoming a tripping hazard.
Imagine how dangerous it would be if the very thing you rely on for support on the stairs is loose? The handrail should be checked to make sure it is not wobbling or loose.
Does the staircase and its surrounding area have good lighting? This is particularly important for those whose eyesight is failing or elderly ones.
Do not polish wooden stairs too much as this makes them treacherous for anyone wearing just socks. Make sure any loose stair carpeting is fixed right away.
By performing some simple checks of the stairs and landing etc. you will go a long way to making sure they do not become a safety hazard.