Sound insulation is a big question to answer for most people looking to finish or remodel their basements. It can seem like an added, unnecessary cost, but the difference it can make is impressive, depending on how you do it.
If you choose to add no sound insulation to your basement, you'll be able to hear pretty much everything happening upstairs when you're in the basement. When you're on the floor above, it will be slightly quieter, but things like TVs and people talking below will be audible. In addition to sounds, people in the basement will hear everything that vibrates, including walking on the floorboards above. Sometimes this banging noise is what's hardest to deal with when in basements.
One of the first things you can do is add carpet to the floors above. It dampens the vibration of people walking and absorbs some of the noise. It's one of the most cost-efficient ways to decrease sound in basements.
Moving into the basement, if you're dealing with an unfinished ceiling, you have quite a few options. First, consider adding 5/8" drywall or thicker. You can also add multiple layers of drywall for double protection. And if you want to add even more, there's a product that some people swear by that you can spread between the two layers of drywall to help with sound even more.
In addition, using insulation in the ceiling joists can help absorb even more sound. You'll lose the benefit of heat traveling up through the floorboards, but the sound barrier often outweighs it. A higher r-value doesn't necessarily mean more sound insulation. Look instead for the thickness that best fills the open space in the joists.
And finally, consider using resilient channels. These are metal strips that you attach perpendicular to the joists, to which you fasten the drywall. This effectively hangs the drywall below the joists so it doesn't touch them directly. This keeps much of the vibration of walking above from reaching the basement. Be careful though, we've heard that 60% of these channels are installed incorrectly and can lose their sound effectiveness with just one screw being driven into the joist, which transfers the vibrations to the drywall, and then to your ears.