Is he who owns an internet connection, and whose siblings use it for illicit purposes without his knowledge, sinful; and is he obligated to prevent them from doing so?

It is impermissible to allow children to use the internet if it is highly probable that they will use it for forbidden purposes, as this constitutes aiding in sin. Allah Almighty says: (And cooperate in righteousness and piety, but do not cooperate in sin and aggression). It is advisable for the father to control internet access to avoid doubt.