How should I finish basement walls with half-height blanket insulation in place?

The blanket's providing almost nothing, so get rid of it (in that placement) and grind its fasteners down to flush with the wall. Research and testing has determined that a rigid board insulation (some sort of foam or rock wool, which requires a vapor barrier) tight against any exterior wall and those joist bays above the wall are the only way to go.

That way, you're minimizing and even eliminating any place for water vapor to collect and warming the exterior wall to avoid water vapor from condensing. Once that convection layer (no less than 1-inch thick) of rigid insulation is on or just rested into place, then the old fluffy insulations can go on top along with a stud wall.

Though, more rigid insulation is most beneficial. Any stud wall, in your non-load bearing situation, can then be comprised of as little as 1"x2" furring type members or regular dimensional lumber laid flat...to save floor space. And actually, you'll get the biggest benefit from insulating the ceiling. Being able to finally hold heat or cold in the room, may make you question whether to finish the walls or to just strip and repaint them...start with the ceiling, you'll be surprised and may be done a whole lot faster and easier (no new electrical requirements to satisfy).