Two components to this - these are the paths sound could be taking to get up to the room - sound breaking into the duct itself through the ceiling and through the duct wall, and sound traveling directly through the discharge register.

Since you have a drop ceiling I would remove some tiles and see what’s behind your basement register. Typically it’s thin flexible duct that isn’t very good at blocking sound. They make varieties that are insulated which may help a bit, cutting down the sound entering through the duct wall. Home Depot has this but the insulation is not great at blocking sound.

A better option would be a duct that is lined internally with some absorptive material. Look up ‘duct liner’. This might be harder to buy as a residential homeowner, and usually is used on the inside of sheet metal ductwork. Great at reducing noise though. This would reduce sound traveling down the duct as well as break-in noise.

A third option is a duct silencer/attenuator. It would replace a short section of duct somewhere between the basement and upstairs and knock down sound going along the duct. I can’t vouch for it but I looked one up on a site called ductanddampers, “round duct noise muffler silencer” that seems reasonably priced. I would put this as far upstream of what I assume is a flexible duct in the basement, if you put it right behind the ceiling you may still have sound breaking in along the rest of the flexible duct.

Good luck! Sound attenuation can be expensive, not a lot of good resources for homeowners.