If this is typical construction you probably have standard 2x wood joists with a plywood or OSB deck covered by upstairs flooring, then drywall directly on underside of the joists, serving as the finished basement ceiling.
Guessing the joist space is minimally insulated? Full insulation in these cavities would help a bunch, but tough to do after the fact.
That paint you are referring to, won't do much.
First I would make sure there are no open gaps between upper and lower where noise is getting through; door seals and thresholds, cut outs around duct, pipes, etc. If so, seal these up first. Further, if you have a cheap hollow core door at basement stairs, lots of sound get through those.
Secondly, you need sound deadening "soft" materials both up and down. Hard drywall walls, wood floors, tile floors, etc. don't absorb sound, they bounce it around making things worse.
Some suggestions that won't break the bank:
Rugs or carpet with pads, upstairs and down.
Add "soft", sound absorbent ceiling tile or panel on underside of basement ceiling.
If you had the headroom, you could install a lay in acoustic ceiling in basement with good sound absorbent tile and another layer of fiberglass insulation on top of it. This will require lighting rework.
Sound deadening "soft" materials on walls; fabric, cloth, etc. -vs- hard drywall, etc. Could try that paint on walls?
Ever been in a Buffalo Wild Wings BW-3's ?
Those places are noisy as heck. All hard surfaces; concrete & tile floors, hard walls, exposed steel deck for ceiling, etc.
SOFT Materials absorb noise.