I don't save my passwords. That is to say, if I log into a site, I don't let my browser remember the password. Neither do I use one of those programs that will generate random passwords and remember them for you, and log you into things automatically.
Two reasons are the core of why I go this route: 1) it forces me to actually remember my passwords, and 2) if I step away from my computer and somehow forget to lock it, unless I'm already logged in somewhere, someone can't just walk up and get to say my bank information.
I don't know, from a security standpoint, if this is ultimately the best way to go, but from the standpoint of logging into a site somewhere else - on my phone, my laptop, the reference desk computer - I don't have to try to guess at most of my passwords.
On computers at work, I go one step further and wipe the history/cookies every time I close a browser. It's also why, despite how convenient it is to log into Chrome and have all my bookmarks and such right there, I won't log into Chrome on a work computer: it's too hard to get your profile information out of Chrome again. Google doesn't recommend signing in on public computers, which is a shame, because that synchronized data is so handy. I make myself remember my passwords, but I bookmark things so I don't have to remember.
And that's probably the biggest reason I'm still using Firefox over Chrome at the moment - if I could sign in and then out of Chrome at work and it wouldn't keep my profile, I'd probably switch. As it is, I've got no reason to. Upside: I usually remember my passwords. Downside: I don't always remember the site to use them at.