I'm going to preface my response by saying that I am 26, I've been playing video games ever since the Sega Master System, so bear that in mind based on my response (which may get long winded).
I think it was easier for someone my age to grow up with minimal parental supervision when it came to games simply because games weren't super realistic. Sure, I had games like choplifter, or my military flight sims, and later Wolfenstein 3D, but all in all, most games where you played as a person just weren't realistic enough to warrant needed parental supervision.
Side note: My father was in Vietnam, and from a very young age I have been exposed to guns, and was given a VERY stern lesson on what a gun is, how it should be used, and how it should not be used. This may have made an impact on my playing of violent games.
Anyway, back on track. Todays games are far more realistic in terms of interactivity, and in my humble opinion, you most certainly should be very aware of what your child is playing. This isn't to say that every time they want to play a game you need to be right there with them, but you should be fully aware of what games they own, and at least attempt to be aware of what games they may have played outside of the home.
Anything you ban them from playing at your house without a very thought out and logical reason for doing so, they will find a way to play. Telling your child, male or female, that they aren't allowed to play GTA4 or whatever just means that they will play it at a friends house when you aren't looking.
Probably the best advice I could offer you is this: Start renting games. Avoid buying games for your child, because I remember for me, as a child, there were very few games that held my interest for more than a week. Especially now that save files are no longer stored within the game itself, this shouldn't cause too much trouble, and you will probably save a lot of money in the long run. Additionally, if you want them playing PC games, download demos for the same reason.
Just make sure you know what they are playing, and that they are aware of the differences between video game reality and real reality.
EDIT:
Addendum - do NOT, under any circumstances, allow them to become a shut-in. I was limited to an hour of video games on school nights. I don't remember if I had limits on the weekends, but it might have been after my homework was done. (This was for elementary school).