Don't know about 1Password, but LastPass encrypts all passwords client-side. Only marginally less safe than storing passwords in an encrypted file on your personal machine.
I can vouch for 1Password as well. Fantastic software, and they have numerous posts detailing their exact encryption techniques. Remember one really good password, and the rest of your passwords can all be completely random. You can sync it over wifi or Dropbox. I don't know the passwords to just about any websites now thanks to 1Password. We mandate it at our company now too. If you can tell me the password to "Important Website X" off the top of your head, you're doing it wrong.
<BTW: This goes without saying, but I'll say it anyway: Don't lose your 1 master password or your password database. ;) >
I use KeePass. It's great. I looked at LastPass but I didn't like that it is "Browser Based". I know it is supposed to be working "offline" but somehow I felt safer using KeePass where I can use the programs own UI, make sure the exe doesn't "phone home" and where I can directly see the encrpted db file and do with that as I please.
At first I worried about what would happen should the db file get corrupted somehow. But then I set up the backup tools with which I can keep as many backups in as many places as I want, so unless the db file gets corrupted in all places, I think I will be fine.
