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.
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.