Some games from Falcom put a little more thought into this. Characters don’t just fall. They bounce.
A good example appears in Legacy of the Wizard for the NES. It’s technically part of Falcom’s Dragon Slayer series, and it sends an adventurous family (and their pet) through a ludicrously complex maze. The gameplay resembles a side-scroller, so the members of the clan can jump, climb ladders, and, of course, fall from great heights.
Observe this example with Pochi, the family’s loyal monster pet. I’m using him because he's a dog in the family portrait and a dino-dragon in the game and therefore the best character.