How well does Apple Remote Desktop work over the Internet?
Apple Remote Desktop will involve either having a VPN or open ports and static IPs. I think you will be much better off with LogMeIn.com's free version, which will require zero intraction from your family members and open ports are not necessary. You will need to know a user's credentials on their computer. If they require more privacy they can disable LogMeIn from the menu bar item, as it will always be on as long as they are connected to the Internet. I suggest adding a user to each computer with your name and one password—that will be one less thing for you to remember and if they change their password or don't remember it you can still help them. 

I use both ARD and LogMeIn daily, but ARD is more than you need.