There is some great commentary on the state and problems of Internet Banking here in the UK: Online Banking In The UK Leaves Customers Short-Changed.

I used to bank with First Direct here in the UK, an arm of the massive HSBC, and they were one of the first to offer a mainstream and comprehensive online banking service. I've been using their online banking service since it was first announced - certainly for five years or more. There are some things to love about the service - for example, they have an integrated SMS service that I can control through my Internet banking interface, so I get a message on my mobile when the balance is getting low, there's been a big payment or deposit, and at regular intervals so I can keep an eye on my finances without having to login. Everything is coded, so it's impossible to tell, even if a stranger had stole my phone, what account the message is referring to.