I find it interesting that carrying her down them seems to trigger the same fear. That makes it seem less like the fear is the stairs themselves (fear of falling, etc.), and more like it is something else. Is she afraid of being picked up in general? Does she have the same response to being carried if it is just around your home and doesn't involve the stairs/going outside?
What is the process of going downstairs like for her when you carry her? Is there a point where she becomes less or more afraid? Could there be one specific scary item on the path? Or is it possible that her fear is of going outside/leaving the safe space of your home?
Here's an experiment to try. Carry her outside and let her do her business, sniff around, give her treats and praise, etc. When she seems calm, start to head back inside. Before you climb any stairs, turn around and walk back away from the house. Is she happy to follow or is there any resistance? What about if you climb one stair with her and then have her turn around and go back down? Two stairs? Three? Is there a specific point where the fear starts to show up? What if you get all the way back to your front door?
When the fear does show itself, have her still go back to the outside and wait until she is calm before taking her back in.
Hopefully, this will help give you a better idea of what the specific problem is.
There are a few ways for you to move forward. I highly recommend you look into a professional dog trainer who specializes in fear and anxiety. Dealing with fear is hard and your pup is young enough that any accidental mishandling of the situation could have life-long effects. A professional would be able to guide the process safely and help you make a plan for solving the problem.
If professional help isn't possible right now, I recommend doing some research on curing fears in dogs. Once you have a clearer idea of what the core of the issue is, it should be easier to come up with a solution.
Best of luck!