I can't believe I'm having to do this (defend Microsoft a little here), but the reality is that Windows has a lot more unknowns to deal with than does Apple. Apple has a distinct advantage in controlling both hardware and software. It is also using an OS that is based on a form of UNIX that has proven fairly stable through decades of international use, fixes and debugging.
So, to say that Windows is less secure is a bit naive. It has many, many more attackers to deal with. In fact, at the last "hackers" symposium, it was shown that Safari was the least secure browser and the first to fall.
The only problem is that the way in which these competitions are setup are inherently in favor of systems like Microsoft. If you give anybody admin rights to login to a system and setup an application to control it remotely you're going to find people that can perform this feat. The real issue is what is the likelihood of that happening? It is like leaving a car with the doors open in the middle of the street saying "try to steal this" and leaving a screw-driver and wire cutters on the driver's seat.
The whole point about OS X is that they make great efforts to keep the identity of the car invisible to begin with... it is like it is in a closed garage, locked, security system activated and lo-jack to boot. Maybe not the most secured, but why bother trying if there are plenty of easier targets?
If you can explain it like that to someone in your family, they may understand it conceptually. It really has little to do with how many systems are out there. Apple just has a much better (imho) way of addressing the issue than does Microsoft. I've been proudly using Macs since 2001 and have NEVER had one virus, trojan horse, or anything of that nature on any of my systems.
But the biggest benefit of all (if you're looking to convince your family of the benefits of a Mac) is very simple; after all is said and done, Apple saves you time. They don't "avoid" problems by bringing out new products every week (i.e. see cell phone manufacturers). They provide a product and continue to refine it long after you've purchased it. The issue isn't viruses/trojans, etc., it is all the time wasted talking/writing about it and fixing/dealing with it that you save... (like reading this long post for instance...)
Have fun!