I'm willing to be persuaded, but I'm currently using two different machines, each with two different browsers open, and this is a relatively light usage case... It's a remarkably complex problem.
1Password can sync between multiple machines via Dropbox. I'm not sure about other OS, but on a Mac there is a browser plugin for Safari, Firefox and Chrome (and a companion iOS/Android app).
I've been using 1Passw[or]d since 2007, and literally all my passwords are uniquely generated (including server root passwords, database root, etc.) At one point I'm a bit scared if I ever lost the database, I'd lost access to all websites forever (because I don't even "know" my email password).
If you're using this 1Password for everything, how do you log in to your Dropbox?
Assuming you have a passphrase for Dropbox, as well; then, I didn't know it had a web ui, and that ~does~ make things convenient --- assuming SSL or similar for security.
I don't use (no longer use) Dropbox. 1Password database is synced to my phone via Wi-Fi, and I always have my phone with me so it never really a problem. If I ever leave my phone elsewhere, then I've got a bigger problem anyway.
Dropbox Web UI is just a HTML implementation of 1Password (read-only) sitting in your disk, so its HTTP security depends onto Dropbox (or whatever sync service you use) rather than 1Password itself.
