All perfect praise be to Allaah, the Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
First of all, may Allaah reward you for being careful not to commit any act of disobedience to Allaah. Fearing Allaah is a characteristic of the believers. Allaah says (what means): {Those who fear their Lord without seeing Him, while they are afraid of the Hour.}[Quran 21:49] So you should always endeavour to be so and ask Allaah for help and guidance.
If you had got the Internet connection while you had intended something evil with it, or if your intention was good and then you disobeyed your Lord afterwards, or if you were the cause for your family committing sins with it, then you must repent, as the one who helps another person to sin is an associate with him in sin. Allaah says (what means): {And cooperate in righteousness and piety, but do not cooperate in sin and aggression. And fear Allaah; indeed, Allaah is severe in penalty.}[Quran 5:2]
Note that a Muslim does not commit Shirk (associating partners with Allaah) just by watching movies that contain Shirk matters, but one commits a sin by doing so.
As regards the steps that you may undertake in this regard, you may do the following:
Firstly: you should try to convince your family in a gentle manner not to use the Internet in matters that involve disobedience to Allaah. If they insist to use it that way, then threaten them to totally cut their Internet connection hoping that they would be deterred.
Secondly: If they do not stop doing so, then you may put a password for using the Internet in a way that no one may have access to it except in your presence, and you should endeavor to prevent them from using it in any religious prohibitions.
Thirdly: If keeping the Internet would definitely lead to them using it in what is forbidden, then it becomes an obligation upon you to completely stop this service [Internet] and not transfer it into another person's name because this is also helping them in sin, if you predominantly think that they would use it in sin. In this case, you should look for another means through which you can get the information that you need for your studies. The same applies to your sister who might need the Internet for her studies.
Finally, it should be mentioned that it is not permissible to be courteous to anyone in what causes the wrath of Allaah. ‘Aa’ishah narrated that the Prophet said: “Whoever seeks Allaah's pleasure at the expense of people’s displeasure, will win Allaah's pleasure and Allaah will cause the people to be pleased with him. And whoever seeks to please the people at the expense of Allaah's displeasure, will win the displeasure of Allaah and Allaah will cause the people to be displeased with him." [Ibn Hibbaan]
Allaah Knows best.