Hey, what Engine are you working in? I'll try to keep it general, my way of implementing it would be:
- Check for climbable objects to the side your character is facing (in a really small distance, you would want to touch the object)
- While a climbable object is nearby, disable gravity (and possibly jumping)
- move the character according to your up/down input
- re-enable gravity and jumping when no climbable object is to the side
- Profit!