Dropbox runs a binary on your machine; that's enough to suspect them. Stick with an open source password manager and an open sync service (S3 plus a script? Or a third party client like Arq).
Yeah, with Dropbox software running on your machine, you not only have to trust them not to snoop on you, you have to trust their non-auditable code to be ~perfect~ against exploitation by others.
With KeePass I haven't felt the need for a browser plugin: Ctrl+C, Ctrl+V is easy enough for my tastes. Plus, in Windows the "auto fill" works more often than not (reducing things to just Ctrl+V in KeePass).
There are multiple KeePass clients on iOS and just about all of them support things like Dropbox sync.
Browser plugins saving me from having to copy/paste are a huge win in my opinion. Prevents me from accidentally copy/pasting things and makes for really nice login behavior.
I just save+sync passwords in Firefox and use a strong master password. I (usually) only need to paste the password from Keepass once unless I elect to not save it (such as with financial logins).
> does it work on ios?
Google seems to return lots of results for iOS Keepass apps. You'll want to vet them on your own. I use KeePassDroid on Android and like it well enough.
I tend to use all 3 of the browsers for slightly different things so having plugins would be ideal but I suppose I could slum it with copy/paste as long as I follow the password/login route to reduce the chance a password gets exposed.
I forgot about my nexus tablet but android is the other thing to have a look into.