Definitely don't keep a long lasting cookie for sites like this. When financial/similar information is at stake, erring on the side of security is good UX as it gives the user peace of mind and they're likely familiar with and understand common security practice. Every bank website worth using automatically logs you out after a fairly brief timeout and there's no "remember me for 2 weeks" option.
A better route to go might be Amazon's pattern where you remember the username of who logged in, but require a password any time the user tries to do anything that would require or access private/financial information (a pattern I explain a bit more about in this post).
Chase.com does a similar thing, slightly more secure:
There's an option to remember one's user ID on this PC. This helps me log in a bit easier but the most secure part of my credentials, the password, still has to be entered each time. Note that unlike Amazon there's no reason you'd be on Chase.com except to view financial information or to make payments. Amazon has it's half-logged in state because some stuff like browsing (and tracking) can be personalized without significant privacy issues, unlike say paying your credit card.
Also as you note, most browsers have an option to remember user names and passwords. I've seen some sites evade those prompts, but I'm not sure if that's by design or just a result of awkward HTML structure making it so the browser doesn't recognize fields as part of a login (I guess it could be both).