Freezing movement on axis - only in one direction?

I can tick a box to freeze movement on the Y axis, but what if I want to be able to move only in direction? (Can go down, but can’t go up)

Without more information is is hard to know exactly what you are after, but depending upon how you are controlling your character here is how I would do it:

public float movementSpeed = 10.0f;

void Update() {
    Vector3 position = transform.position;

    float vertical = Input.GetAxis("Vertical");
    if (vertical < 0.0f)
        position.y += vertical * movementSpeed;

    transform.position = position;

The player character moves on a flat surface. I don’t want it to be bounced up when it collides with objects - which happens sometimes, but I also want it to fall off the stage if he goes over one of the edges.

I hope this clarifies what I have in mind.

If you want it it to fall off an edge you can add a rigidbody to the gameobject and play with the settings bid.
Hope that helpt a bid.