dnd 5e – What ability or skill check would cover surfing on a shield down stairs, and what would the DC be for such an action?

Acrobatics (PHB 176 – emphasis mine)

Your Dexterity (Acrobatics) check covers
your attempt to stay on your feet in a tricky situation, such as when you’re trying to run across a sheet of ice, balance on a tightrope, or stay upright on a rocking ship’s deck. The DM might also call for a Dexterity (Acrobatics) check to see if you can perform acrobatic stunts, including dives, rolls, somersaults, and flips.

Note that if all four of them are attempting to stay on a single shield, this would be a group check.

Group Checks When a number of individuals are trying to accomplish something as a group, the DM might ask for a group ability check. In such a situation, the characters who are skilled at a particular task help cover those who aren’t. To make a group ability check, everyone in the group makes the ability check. If at least half the group succeeds, the whole group succeeds. Otherwise, the group fails. Group checks don’t come up very often, and they’re most useful when all the characters succeed or fail as a group.

Finally, see:
What guidelines if any exist to guide the DM in establishing difficulty level?

You know more precise information that should factor into your decision – Polished marble steps? Unevenly worn stone? Carpeted? Are the stairs particularly steep or shallow? What is lighting like? etc. You also know the tone of your campaign – gritty realism or romantic fantasy?

Based on your description, I would likely assign this as somewhere between Moderate and Hard. Unlike the other two points, though, that is not a rule but rather a ruling, and it says as much about me as a DM as it does about the situation itself.

You should choose a number that reflects your assessment of how difficult this task is for these characters in this situation in your world.