Use the PHB's rules for Jumping
I think the best way to address this is to just focus on the PHB's rules for jumping that we are already given. They do a pretty good job of addressing this situation.
Your Strength determines how far you can jump.
Long Jump. When you make a long jump you cover a number of feet up to your Strength score if you move at least 10 feet on foot immediately before the jump. When you make a standing long jump, you can leap only half that distance.
So we have two situations, one where the character could make the jump under normal circumstances and one where they could not. You say they are able to jump across without the obstacle so I will assume their Strength score exceeds the distance to be jumped.
This rule assumes that the height of your jump doesn't matter, such as a jump across a stream or chasm. At your DM's option, you must succeed on a DC 10 Strength (Athletics) check to clear a low obstacle [...], such as a hedge or low wall.
We already have a rule for requiring an Athletics check when you need to clear an obstacle. I would consider pushing up against a waterfall to be equivalent to leaping over a low wall, though maybe slightly harder. Therefore a DC15 Strength (Athletics) check to make the leap is a good idea.
When you land in difficult terrain, you must succeed on a DC 10 Dexterity (Acrobatics) check to land on your feet. Otherwise, you land prone.
I would consider the wet rocky ground behind a waterfall to be difficult terrain, therefore the DC 10 Dexterity (Acrobatics) check to land on their feet is required. If you like, place disadvantage on this check because they can't see the surface they are landing on, I would rule this way.
According to the rules and your description the jump itself is fairly simple, except for the waterfall and unseen landing. Therefore I suggest the following checks:
- DC15 Strength (Athletics) Check to make the jump
- On a fail they fall into the water.
- DC10 Dexterity (Acrobatics) Check at disadvantage to land the jump
- On a fail they land prone.