I don't know of a specific bank, but I'll make an argument for entering the data in manually from your statement (paper or otherwise): It forces you to read your statement in detail and increases the likelihood of identifying an erroneous/fraudulent charge. It may seem like a lot of work, but I really only spend about an hour a month for 2 credit cards, 2 checking accounts, 2 IRAs, and one savings account. I also think it feeds my subconscious with a subliminal reminder of how much I'm spending each month. When I see several charges for dining out, for example, it makes me want to pull up a spending report and verify that the trend isn't headed up too steeply.
Read my 'fiscal fitness' financial disclosures here.