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.