Regarding what was mentioned three posts above, I think thats the best way for a brother or sister to save themselves from falling into any problems, by letting their parents know passwords etc. Its correct, there should be absolutely nothing to hide. I think that if a parent is not concerned with what their children are doing on the internet, then this is negligance on their part. The internet is a very diverse place, but unfortunately there is clearly more evil than good.
One of the taabi'een said: If one can take all his feelings and all his thoughts and place it on a platter, and then walk around the marketplace for everyone to see, and have no shame at all about anything on this platter, then this is taqwaa.
If one were to keep this definition of taqwaa in their minds at all times and act accordingly, then the validity of notions such as a brother chatting with a sister 'discussing religious matters' is suddenly rendered null. If a sister is afraid of her brother, husband or father reading something she writes to a non-mahram male, let it be in regards to deen, then who has the more right to be feared? As it comes in the Qur'aan, Wallaahu a'haqqu an takhsaah; Allah has the most right to be feared.
As far as monitoring cell phone (mobile phone) activities, that is also very important. Laa hawla wa laa quwwata illaa billaah, i've heard stories of married sisters having conversations with random guys through text messaging, and almost being picked up by those guys. Had their husbands been occasionally keeping a check on the call logs/text msg's, this would have not started from the beginning.