I'm homebrewing a rotten, collapsible staircase. I want there to be a Dex check to climb it, but if they fail the check, they can make a save to avoid falling.
Should the check or the save be of a higher DC?
For saves and checks in general, is one typically higher than the other?
There is no rule about which is harder
DMG p. 239 says the following regarding setting the Difficulty Class (DC):
(The table also appears here in the basic rules.)
In your given example, think about how bad you want the stairs to be. Are they ready to collapse or only a few bad steps? On a fail, does only one step give way or the whole flight? Etc.
A sample scenario
There is a flight of rotting stairs. 5 steps are too weak to support any weight. A DC 15 Wisdom (Perception) check is required to spot the rotten steps. On a failed check, the PC needs to make a DC 12 Dexterity saving throw to avoid falling through the rotten step; a creature takes 1d4 bludgeoning damage on a failed save.
NOTE: Depending on the character's ability score and proficiency, a DC10-DC15 usually has a 50% chance of success.