I've recently switched from a web based password manager I made myself to KeePass, and I'm quite satisfied so far. The encrypted database can be kept locally, but if you're OK with keeping the encrypted file on Dropbox (or Google Drive etc.), it allows for seamless synchronisation with Keepass2Android. It's been working pretty well for me. Of course I can't test whether you'll be able to access it on your corporate network, but Keepass2Android keeps a local cache of the database, so you can synchronise when you have a network connection and call up the password list whenever you need it, even without a connection. It can even merge databases if they get out of sync due to concurrent edits.

As for security, KeePass uses a configurable number of iterations of AES (I think the default is 6000 or so) to encrypt the entire database, so I think that point is covered.