Disclaimer: The solutions we've shared are just one exciting approach, and there are surely many other wonderful methods out there. We’d love to hear your alternative solutions in the community thread below, so let's keep the creativity flowing!

$9$th step on the first ladder and $13$th step on the second ladder.

On the $k$th step of the first ladder, the distance from the ground is $\frac{20}{34}\cdot k$ feet.

and on the $m$th step of the first ladder, the distance from the ground is $\frac{20}{49}\cdot m$ feet.

The distance between them is $| \frac{20}{34}\cdot k-\frac{20}{49}\cdot m|=\left(\frac{20}{34\cdot 49}\right)|\left(49k-34m\right)|$.

This is minimized if $|\left(49k-34m\right)|=1$, which is true when $k=9$ and $m=13$.