Frame around pipes that run on outside walls by building the wall around them. Lay the bottom plate for the wall on the floor in front of the pipes, and mark where each pipe is. Cut a notch out of the bottom plate in each pipe location, deep enough that the bottom plate can slide around the pipes and meet the outside wall behind them. Alternatively, if you have a big basement and aren't worried about space, you can simply frame the wall in front of the pipes, leaving them hidden between the framed wall and the exterior foundation wall.
Build a special framing to accommodate pipes that run in awkward positions that can't be hidden within a wall. Cover the framing with drywall to make the pipes invisible. Use 2-by-3 lumber, which is sufficient for framing that is solely for surrounding a pipe, as this framing isn't load bearing.
Move pipes that run across the floor by re-plumbing them across the ceiling joists and then directly down to wherever they're going. Frame a wooden raised floor in that part of the basement to accommodate the pipe, as an alternative method.
Build a wider wall out of 2-by-6 lumber to avoid making many custom build-outs for large pipes. Position the 2-by-6 wall so that large pipes fall within the framing and don't interfere with the plane of the drywall on the sides of the wall.