I just bumped up the "paperwork" post I made a few months ago for you to have a look at.
In other words, my answer to your question is basically within that post ... in order to make my family feel better about my touring, I take all the precautions I can with regards to things like insurance, travel plans, financial issues, etc. etc. etc. I carry copies of the paperwork I need for those things with me, and I leave a copy with my family. That way my family knows that if something goes wrong, I will have the insurance coverage to deal with it, and because they know my approximate itinerary, they have an approximate starting point to look for me, if necessary.
I also try to contact my family on a fairly regular basis. In Australia computers and internet cafes etc. practically grew on trees. Here in Canada, it is also fairly easy to find places with internet access. In the US it was a bit more difficult, but still do-able. So, while I am on tour, I fire off emails whenever I come across a free internet connection. That lets my family know that I'm OK, and also lets them know where I am and what I am doing.
I also try to travel with people (at least one other person) I know as much as possible, and/or to stay with family or friends along the way. That gives me a sense of comfort as much as it does my family.
My family is so happy that I'm doing more bicycle touring and less motorcycle touring this year that I haven't heard a peep about the dangers of long-distance bicycling. I think they believe it's something harmless that kids do because they can't get a driver's license yet. One of my brothers is the exception - he also rides, and we're planning a week-long tour this summer. But he also rides a motorcycle, so the family is just relieved we're off the "donorcycles", as they call them, and on to harmless kid stuff like bicycles.