There are two things you need to consider when soundproofing basement. First is the elimination of all cracks including the tiny ones. Second is beefing up the sound resistance of all walls and ceilings. It's hard to beat drywall (sheetrock) as a surface covering on both these concerns. Finishing basement with drywall’s ability to absorb sound is thanks to its density. Drywall tape and joint compound seal the sound-transmitting joint cracks. The best type of drywall for basement ceilings and walls is 5/8"-thick fire-code material. The gypsum core of the commercial grade includes glass fibres for extra strength, besides being thicker than the regular 1/2" material.
Beefing Up the basement ceilings makes sense since most of the sound that travels up from a basement comes through the floor. When soundproofing basement it makes sense to focus your greatest attention there, at least initially. Simply putting 1/2 drywall on the underside of the floor joists boosts STC (Sound Transmission Class) of the whole floor structure to only about 36, which is not nearly enough. A much better plan is to install 5/8 drywall onto the bottom side of the floor joists, along with sheet metal strips call resilient channel. These channels separate the drywall from the floor joists, which blocks telegraphing sound transmission more effectively than direct drywall-to- joist contact. By adding glass fibre battens between the joists before drywall installation you will boost the entire floor to a STC of over 50. Finishing basement this way is what you should be doing anyway.
Finishing basement ceilings is a good start at soundproofing basement areas but do not forget the doors. The usual cheap interior, hollow door transmits sound very well so replace it with a exterior grade solid core door. Also you should enclose the furnace or AC unit in a room of its own since sound will travel through the ducting. Finishing basement for soundproofing is great but you may wish to carpet the first floor as carpet absorbs sound very well.