I wouldn't. Is the email address genuine, i.e.the one they usually send you mail from? Are there any links in the email? If so, hover over them, where they go will appear in the bar at the bottom left of the window. If it is where you expect, then ok, but if it's rubbish or somewhere strange, then ignore it.
Hardly anyone will ask you for that information by email, particularly passwords. even passwords over the phone only ask for specific letters, not the whole password.
I say ignore it.