I have a PIII 650MHz, 10G HDD, 128MB RAM and intend to use it for dial-up internet. If I upgrade it to 384MB RAM would I achieve satisfactory browsing speed?

The RAM upgrade does not really affect the browsing speed as much as the speed of your internet's ISP. The RAM upgrade only boosts the performance of programs on your computer (like MS Word, Outlook, etc), including how fast they launch, and so on.
One way your browsing can be slow is when you visit websites that include a lot of advertising banners and/or popup ads. To avoid all that, and enjoy faster browsing, download Firefox from www.mozilla.org, which gives you the option to block popups. You can then install an extension called "AdBlock Plus" which blocks most banners and such, so that webpages load faster.
Hope this helps.

Don't even think of using Firefox if you don't have at least 256mb of RAM!! RAM will make your browser faster.

Yes I forgot that Firefox is a memory hog indeed. I'd still rather use firefox on a slow PC and block all those banner ads, than use Internet Explorer and endure all those flashy, annoying banners.
Actually, I believe that increasing the RAM only makes the browser load (and shut down) faster, but doesn't improve the actual internet speed.

Swapping to hard disk is makes a memory-constrained computer very slow. Just install the RAM, you won't regret it!