Linux lasts longer, at least according to this CNet article. Unpatched systems on the internet last for two months in the case of Linux, 4 minutes in the case of Windows. I know which one I’d be able to patch before getting compromised.
All FUD aside, Linux is much better for an “always-on” connection, but it still isn’t good enough. My advice: put both systems behind a firewalled router and close all ports except the ones you absolutely need. In the case of your average home user, there will be no servers inside of the firewall, so all ports can remain closed up tight. If you decide to run a server, just forward that port to the server machine. In my house, I run an email server, a web server and DNS. I have only the necessary ports open, and they connect to a Linux machine running all of the servers.
Firewalled routers are cheap these days, I got a 4 port router with 802.11 b/g wireless support for about $70, and that was 6 months ago. The firewall is worth every penny, although the wireless was my real reason for the purchase.
Once you get the firewall set up, install Firefox on all systems. Mozilla would also be OK, and it is a fine browser, but it is a bit too unweildy for grandma, or evn a teenage girl to handle. Whether you believe MSIE gets hacked because it is more buggy or because it is more popular is irrelevant… Switch to Firefox anyway. Then set Firefox to block all popups. That will end all of your problems with spyware right then and there.
Email is a different problem. Outlook, Thunderbird, whatever email program you use, will still have lots of problems catching any virus that comes in. Email vulnerabilities can be manageable if you follow a few simple rules. Never turn on previews. Never open anything that you aren’t sure of. If you cant trust yourself (and you probably can’t), buy something to scan your email. Norton, McAfee, whatever.
I’m going on 5 years of having an “always on” connection, and I haven’t lost any data or gained any viruses yet, despite having 2 teenage daughters online a majority of the time, and my network server online 24/7. Spyware and spam are a different story… I’ve gotten my share of those 
– end of fatherly advice