My family has been Internet banking since early 2000. We moved to an Internet only bank (no physical banks anywhere that we can walk in to) because we were ready to start paying bills online and get our account information online. At the time, our large overhead bank, Wells Fargo, actually charged customers $9.95 per month to have the privilege of online banking. After some research, we settled on an Internet only bank with no physical banks, just a web site, great customer service, high interest rates, and free online banking. Their site has changed only slightly over the years and we have never regretted the decision.
Internet banking isn't for everyone though and that's why brick and mortar banks still exist and are building new branches every day. If you are curious about Internet banking, look through the consumer reviews of Internet banks at Epinions . Also, the Motley Fool has a great overview article about Internet banks as well.