I am in the unique position of actually having experience with this.
My company had just bought a 2 gbps connection sometime back. This was when P4's were newly launched. My PC at that time was a PIII 800 MHz with 64 MB ram and a 5400 rpm HDD.
Anytime my download speed crossed approx 750 KBps , my pc would hang until the download either slowed down (rarely happened, the servers being used were really fast) or until the download was over. Increasing the ram to 128 MB allowed me to keep working uptill speeds of around 2500 KBps, which I almost never reached, so there was a drastic difference in usability of my computer.

I think I got off the point a little, other than extreme cases where the network hardware is real bad or there are i/p problems with the pc, ram makes no appreciable difference to the browsing speed.

No modern pc will have show any difference in browsing speed with more ram installed.