If the computer is that old, then the user is an outlier and not the target audience for the modern major browsers. Better off finding a niche browser.
Actually my computer is only four years old, and supports up to 2GB of RAM.
It is hard to find a working niche browser when Mozilla and Google are doing everything to make sure the web depends on their behemoths' complexity. They are pushing for the Web Platform. They are constantly adding new features.
I used to use Links 2 as my primary browser until about a year or two ago. I would have preferred to keep using it (actually I still do, but it's no longer primary), but one day, a few years ago, Google said let's kill IE6. I can't remember the date. But I know web devs had a party and since that day, the amount of Javascript on the web has skyrocketed, and it's gone from an optional "experience enhancer" to a requirement in many cases. So I'm not given much choice.
But this isn't just about me personally; I could buy a new computer today if I wanted to. What people have pointed out elsewhere in the comments is true -- there are people who still have old computers, much older than mine. And people (parents too) who cannot afford much else besides aging second hand machines.
What irks me the most about this whole "web must move forward" mentality is that machines that would otherwise be perfectly usable for the sorts of basic tasks one used to be able to do even without these new web technologies (i.e. read email, news, etc.) are being obsoleted and scrapped by people buying newer, much faster hardware --- only to be used just for those same basic tasks again.
Is that really a bad thing though? These old PC's are extremely wasteful with electricity relative to newer ones, so how much money are people really saving? People are so quick to ignore the ongoing costs and obsess over the immediate.
Basic web browsing/word processing is incredibly cheap to do right now in a new PC (sub $200 sans peripherals for the absolute newest hardware). In my opinion, these are machines that will handle web tasks for at least the next 10 years barring some huge shift that I cannot foresee.
Why do you torture yourself like that when another 4 GB costs $15?