Its only going to pick whats in front if you shoot the ray from the camera forward in the look direct, you want to pick whats below the player then shoot the ray down.
Physics systems like bullet or nvidias physx have character controllers that auto step up a given height, making things like this a lot easier, thats what I would suggest using, unless you are trying to do it yourself that is.

Just to try and clear up some confusion...

What was suggested is that you shoot a ray from your camera's position downward and find the ground.

This isn't done through picking (the process where you find whats under the mouse on the screen), this is done through what's called a ray cast.

Basically you test a ray for collisions against your world geometry.

It doesn't have anything to do with what is or is not on screen or under the mouse.

It has more to do with what is under the player's feet and how far it is to the ground from the player's current position.

Also if wondering, often times even though the stairs look like steps, the collision object for them in games is often a ramp. That makes the camera go smoothly up it and simplifies some other things (esp navigation mesh pathfinding type things).

If you go the route of actually having stairs in the collision geometry, you will probably want to smooth the camera movement to make it not look jerky as you go up and down steps. One way to do this is to interpolate to the camera's target position over time instead of teleporting it there each frame.

HTH!