Am I sinful if I provide internet to the house, knowing that my family members might use it to listen to songs and other things, despite my having advised and forbidden them from doing so?

If you know that your household members will use the internet for illicit purposes, and they will only connect it upon your request, then your request falls under aiding in sin, and you will bear a share of their misuse. However, if this is merely a possibility, or if they would subscribe to the service regardless, then your subscription does not constitute aiding them in sin. Piety dictates that one should only subscribe with those whom one predominantly believes to be God-fearing.