Mint asked for bank passwords, and had 5m registered users as of 2011. They sold for $170 million. I believe Coinbase is copying common UX patterns which have been proven to be palatable with users.
Coinbase is neither Mint nor Paypal. We have no idea if they will survive 6 months, and we can't trust they don't store credentials on their servers, even though they say they do. Have they been security audited? Or do we just take a one-line comment on Hacker News as the truth?
If they reach a level of legitimacy, then maybe it might fly with some users. But I personally think it's way too premature to ask for usernames and passwords for banks, especially given how much fraud, hacking, security problems, and monetary loss associated with Bitcoin companies.
I'm not saying this company is a scam, but if I wanted to create an elaborate phishing scam, this is exactly the type of setup I would create.
Hopefully no one is stupid enough to give this key online banking information to this company or anyone else that asks for it.