My Dad's old Windows 98 laptop got thoroughly infested, so in the end I replaced it with a Mini Mac since (i) a half decent cheap flat screen monitor is much much better to see than a laptop screen (important when their vision is getting a bit poor) and (ii) OSX is much less prone to getting virus etc, admittedly mainly because it's much less targeted. Having said that, if they haven't got a permanent location for it, then a laptop is probably a better idea.
Windows has got a lot better on the security front, but I'd still have to make sure that any Windows box had some sort of anti-virus and/or anti-malware software on it.
Wireless has a lot going for it, it saves having wires around that can be tripped over or jammed in doors. You can put the router under a table next to the phone, or ideally screwed to the wall, and it's out of the way, and safe from damage. Modern wireless routers are pretty reliable, they rarely loose the connection. The worst case scenario is that they need to power cycle it, and give it a couple of minutes to reboot. Since this is almost normal behaviour with Windows problems, I can't imagine it'll be that difficult to get used to.
If you've got the router that is recommended by your ISP, it'll probably be fairly reliable on that connection, otherwise they'd have thousands of support calls to deal with.