how to soundproof pipes in basement apt
Supposedly (per hazy memory of TV show) cast iron pipes are quieter. Lacking a TV show's need to tear things out to make footage, and budget for silliness, I'd probably skip tearing out plastic pipes to replace with cast iron in the hopes that it would be quieter pipe...

Box and insulate. If you want a better job, then box and insulate the first box. Or just stop after the first box, flush, shower, and drain the tub a few times while someone listens below, and see if you feel the need for additional boxing. If you have the freedom of space layout, pencil a closet in around the pipe box, and that's another box between the pipes and living space. 5/8 drywall (because it's dense/heavy/good for blocking sound) and 2x2 in the corners make a nice pipe chase, and you pack that full of cellulose or fiberglass insulation. If it needs more, use resilient channel to attach another layer of drywall, and perhaps add a sheet product like mass loaded vinyl between the drywall layers as well.