It may have a slight effect ie speeding up the processes that send the data. Also, an older PC (with, say 64MB) will have older and less efficient technology (inc modem) than a brand new laptop with 480MB.
If you are using you swap file heavily, adding RAM would help you get faster internet by speeding up the applications. But as it has been said before here, it really depends on the conection you have. Did you old PC have a slower modem???