The PO did a poor job of mudding (don't think there's any tape.) This is in the basement of my bi-level. There is a quarry nearby and there is a lot of vibrations from blasting, hence the cracks, but the upstairs is fine. I know it would probably be easier to just replace it, but it would be more cost effective to repair it. Could I just sand it down and re-tape and mud? What type of sander would I use? Do I have to get all of the existing mud off? Anything else to consider? This is on every seam in the basement.
Sand it with a pad and sanding cloth. It's sold in the same area as all the other drywall supplys. It looks like screening but has an abrasive grit on it. It does not load up like sand paper does.
Get it smooth all over, apply new tape in a bed of drywall compound, then another coat over it. Always use thin coats. For the first coat use a 6" wide knife. Do not snad before the second coat, just go over it with the drywall knife to knock off the high places. apply another thin coat this time with an 8" knife.
Now sand the wall with fine screen cloth using the pad in a ciruler motion.
If there's any low spots or air holes add a tiny amount of water to the compound to thin it and go over the flaws.
Primer and paint will not fix flaw.
DO NOT USE A PRIMER AND PAINT IN ONE FOR ANYTHING!!
Use a primer like Zinzeer 123, look for any missed flaws and fix them, then two coats of paint.
I agree with joecaption. If you sand it down and tape that seam you should be good. If you try to save some time and just mud the crack without taping it the crack will come back. I had a similar situation where a crack kept coming back near the top of a door jam. I was lazy the first time I "fixed it" and didn't use tape. The crack came back but after I took the time to tape it, it disappeared for good.