Basement half wall advice
Man you can make that look really nice.   I would pop a 1x of pine on that and stain and poly it before install.   It only needs a couple of screws (or finishing nails - debate debate!).   And then you have this nice focal point ledge on the basement.   Can hold a vase, the remotes, or just stack it with junk!
I dont know how long of a run you have but - you have a LOT of options.

just the nice pine ledge (pine takes moisture really well, doesn't swell, and the stain doesn't fade fast with moisture)... ok here is the deal.  cool as hell if you can get something that spans the area or if you have an idea to cover the joint of 2 different 1xs with some ornamental wood (or functional)
pop in wall shelving all the way across.
be eclectic - different size pine boards... its a wall.   Different size shelves and whatever.

Do not hide this edge with drywall/mud.   I have a feeling just looking at it + your description you could drywall/mud but that's trying to hide it.   Make it look like you framed it for a purpose.   Make people who enter your basement think - wow wonder why he framed a ledge out like that, love that idea.   I always try to take any weird architectural iffyness and turn it into a focal point.