The job of covering the posts is least difficult if they are made from 4-by 4-, 6- or 8-inch lumber. You simply cut strips from a full sheet of drywall that extend from the ceiling to the floor and screw them to the posts. Cover the wide sides first, and then the narrow ones. The drywall shouldn't overlap at the corners because corner beading -- which is what you use to finish the corners -- fits better if there is a 1/2-inch gap on the edges. If the basement has a high ceiling, you may need two pieces of drywall to cover each post face.You then tape the joint, and mud over it.
You can choose from metal or plastic 90-degree corner beading, and it can be the type you attach with fasteners or it may be self-adhesive. You can also install bullnose beading to give the posts a more elegant, curved look. After cutting the beading with a hacksaw, you either attach it with drywall screws or nails, or you lay an undercoat of mud and apply it like drywall tape -- depending on the type you have. If you have the self-adhesive variety, spray the adhesive on the corners and press the beading flat.
To get the smoothest finish on posts, you start with a layer of mud that flattens the indentations and edges of the corner beading, but this layer shouldn't be too thick, or it will crack as it dries. Several thin coats give better results than a few thick ones. Apply the first coat with a 4-inch knife, but use an 8-inch knife for the topcoats so you can avoid making a seam down the middle of the post with the edge of the knife. Once the final coat dries, sanding it with 120-grit sandpaper and priming it prepares it for texture and paint.
The procedure for drywalling cylindrical posts is the same as for rectangular ones, but first you need to construct a frame around each post. An easy way to do this is cut four strips of 1/2-inch plywood that extend from the ceiling to the floor and screw them together. At least one of the strips should be anchored to the ceiling and floor with metal corner brackets or pieces of wood glued and screwed to the inside edges. You can also construct the frame with 2-by-2- lumber or two-by-fours. Once the frame is secure, it will provide the backing you need to attach the drywall.