General info

1. The rocks are not random, but are based on the timer that starts running as soon as you gain control on this screen.

Incidentally, this is true for every screen of the game. Nothing is dependent on the global timer, everything is governed by a local timer.

2. The RNG changes once every few frames, so there's no consistent way of manipping the later parts of the climb (that I know of).

3. As soon as the screen fades in, hold Right+Up. This will allow you to at least manipulate the starting ladder.

The wrong way to start the climb

Technically, the quickest way to start climbing is to release Right as soon as you touch the ladder. This will snap you onto the ladder instantly.

HOWEVER.

Doing so ensures that you'll get knocked down by a small rock falling directly over the ladder.

The correct way to start the climb

Let go of Right as soon as you pass by the midpoint of the ladder.

When your legs reach the horizontal line in the picture (the bottom of the small dark bit to the right of the ladder), hold Right+Jump.

This will ensure good rock RNG for the next few seconds, letting you climb the first two ladders safely (provided you don't hesitate too long).