I'd still update the browser to the latest version - with you, it is IE 6.x - as there is some software that won't run without.
Regarding the browser, ALL browsers you can install are unsafe, so security-wise it's just a question of risk.
Many pages won't accept old browsers, as they don't provide a secure version of SSL (read "https"), that is, at least TLS 1.0
On W9x, the newest browser is neither firefox nor IE nor chrome, but Opera 10.63, from 10/10/2010
You might need a newer version of the MSIMG32.DLL, though, which you can get from Microsoft directly, i think. If not, PM me.
You can still get that version from i.e. on Toms Guide:
http://downloads.tomsguide.com/Opera,0301-24290-18651.html
It is not only the newest, but also the most capable, fastest, least subject to malware, and doesn't requre any add-ons like kernerl-ex. But that's just my very personal opinion, i don't claim to be the internet-wise-guy.
Cheers,
Ragnar G.D.