IF support staff for a service you are using need access to your account or information within it, they can get that access without needing your password. Nobody should be asking for your password.
This applies to email (e.g. Hotmail/Windows Live, Yahoo, Google and many others), social networking (Facebook, LinkedIn, MySpace, etc.), online photos (Flickr, etc.), and especially applies to your banking and finances. NO bank or financial services employee should ever ask for your password – bank policies generally prohibit them from doing so as a firing offense.
Think of someone asking for your password the same way you’d think about a stranger walking up to you on the street and saying “Hi, I’m with the village. I need your home address and your house keys.” No matter how friendly and professional looking, would you just give your keys to a stranger like that?