pds41: Is that... Windows XP?...As a side point, unless you've got no other option, I'd really recommend running XP air-gapped to protect your system and thoroughly virus check any optical/USB media that you use to load software onto the system.
Yes it is Windows XP, and I've encountered the same problem (starting about a month ago). GOG has changed the scripting on the account page so it no longer works on older browsers such as Firefox 52.7.3 ESR (the last version to officially support XP). GOG support's reaction (a month after I raised my ticket) was a disappointing (and rather predictable) "Please use a computer with a modern operating system for online browsing and purchases.".
Bstone, your best solution is to install and use the latest build of Roytam1's
New Moon browser (a fork of Pale Moon, itself a fork of Firefox). This works on XP and NewMoon version 28.10.4a1 (mentioned in
this post) works with GOG's account page on my system (New Moon version 27 didn't so best avoid this).
New Moon is also considerably faster than Firefox and keeps the "classic" Firefox UI, making it more pleasant to use in my view. In addition you should add plugins to control webpage scripting and I would strongly recommend
eMatrix (4.4.9 being the current version at time of posting).
There is a more general point here, that GOG doesn't see browser compatibility as a serious issue, and given their ever more arcane scripting, other browsers (and other OSes) are likely to encounter a similar issue in the future (i.e. future access to your games library is no longer guaranteed).
As for security, Windows XP can be effectively secured (I'm typing this post on a laptop that was set up with WinXP back in 2007 - a recent check with Kaspersky's offline scanner turned up only one result which was legitimate software). Any version of Windows (or Linux for that matter) is breakable in its default setup - (32-bit) XP's strength (compared to later versions of Windows) is the ability to remove (courtesy of nLite/XPlite) the most vulnerable applications (IE, Outlook, MSN Messenger, etc) and to use security software that patches the Windows kernel (such as Process Guard, System Safety Monitor, AppDefend or EQSecure) which would be blocked in 64-bit versions. If you haven't already done this BStone (and the screenshot of your System Tray suggests not), definitely check those programs out, though I think only SSM is available in its full version now and only via the Internet Archive sadly - for more info cut-and-paste the following URL (this forum screws up Internet Archive links):
web.archive.org/web/20090118125809/www.syssafety.com
Since almost all malware requires Internet access to function, I'd strongly advise installing a
top quality firewall (Windows' own firewall is completely inadequate at blocking malware).