Insulating and installing drywall in your basement may help with temperature control and aesthetics, but it won't effectively soundproof the space. Soundproofing requires additional measures such as using soundproof materials like acoustic insulation and mass-loaded vinyl. Simply relying on insulation and drywall won't block out noise from the outside or prevent sound from traveling between rooms. Therefore, it is important to take proper soundproofing measures if you truly want to reduce noise in your basement.