First question is what’s under the carpet??
Second, id probably redo the rail and spindles. Add a light. If there are hardwoods. AWESOME!! Stain and add a runner. If not I guess keep the Carpet. Maybe add wallpaper

I should have added. It rlly doesn’t need anything. It’s a very nice. Well kept space.

It depends on what you are going for. Want something homey, add large pictures of your family. this way it doesn't clutter the visual space as its the width of a hallway. Want something more modern: try peel and stick wallpaper. it doesn't have to be crazy bold, it can be textural to warm it up and give it a little something with it being a large statement.

Personally, I would not add mirror as it would be reflecting a wall.

Thank you all for your comments!!
To answer your question, there is no hardwood under the carpet.

I would love to redo the handrail and railing. What do you think of these very different options? I like them equally.

Wallpaper is a great idea. Where do you suggest I begin and end the wallpaper?

I have a stairway like yours and I have three coordinating square wall art pieces on the high wall.

I like the first and second railing ideas. For wallpaper- I always say the more the better... you can do the top half of the walls (leaving the wainscoting) and go all the way down. Or obviously continue it to Wtvr is down there lol.