I've seen this from the other side - some of my best friends went through a nightmare scenario last year. Bob was in a serious motorcycle accident and was in the ICU for several months, but Alice couldn't get to ANYTHING "official" without Bob's passwords. There was a master list, but it was stored on the laptop Bob had in his backpack during the crash
Even once Bob got out of the ICU, he still had severe mental and cognitive problems and wasn't able to do anything computer-related for a long time - and still doesn't remember some of the common passwords he's been using his whole life.
Bob and Alice are both techie-type people, and as a result there were a LOT of things Alice couldn't straighten out until she got the legal power of attorney or whatever, which took almost six months. Bob is now back at home, more than a year after his accident, but he still has lots of memory issues and may never recover them all. It's a really scary situation. They did have the major stuff in place (legal will, etc.) but that doesn't help when you're on the phone with a service provider in a different state who is sending you a recurring bill each month and you can't get them to cancel the service because your name isn't on the account
So partly because of that, DH and I made a bigger conscious attempt to make sure we know each other's passwords. I know his basic ones (the ones he uses for everything) and he knows the formula I derive my passwords from. I've never logged into his email, but I suspect I probably could if I sat down and made a list of every password I know he's ever used and worked from there. He's let me log into his World of Warcraft account, so he does trust me
I do all the finances, but I also intentionally get paper statements sent to the house so if something happened (to me, the computer, the Internet, the bank, or whatever) we would have a way to get the information we need right away.
More importantly, we both know the general gist of what the other is involved with online - he knows about eHell and my usual online activities, I know about his gaming sites and programming blogs. We've also had the talk about Scrabble-related sites and are both comfortable with the current level of each other's Scrabble-related Internet time even though we don't discuss the details
(I recognize that this tolerance varies from "none at all" to "go for it" among different couples, but I do think it's important to at least be honest about it with each other!)