How should I handle this thickness mismatch on a basement wall?

First, check the actual thickness of each layer and why the upper section was built that way. If it’s about 1/2 inch thicker, a wide coat of joint compound won’t make a good fix. You could bring the lower section out to the same plane with an additional drywall layer, or use furring strips on the framing to set the new board forward. Make sure the edges have solid backing and the finished surfaces line up.

Before closing the wall, confirm the water source is dealt with and the cavity and framing are dry. Mold-resistant drywall doesn’t replace fixing moisture problems.

If the mismatch is really only 1/8 inch, it’s manageable. You can feather joint compound over a fairly wide area at the seam so the change in thickness is gradual. It takes a few coats and careful sanding, though, and the wider the transition, the less noticeable it tends to be.

That’s a different situation from a 1/2-inch step. I wouldn’t try to hide that much with mud. Also double-check the new sheet’s thickness before cutting it. If it’s the wrong size, exchanging it is likely less frustrating than building around it.

A chair rail or other trim can make the change look intentional, especially if the lower section stops at the same height all around the room. It won’t solve an uneven wall behind it, so check that the boards are supported and firmly attached first.

If you want a plain wall with no trim, matching the thickness or bringing the framing out is the neater route. For just 1/8 inch, feathering the seam is another option.