This is very close to how LastPass works under the hood. You're storing an encrypted database on their service (just like DropBox in your case). They don't actually store your original master password.
The only legitimate security gripe I've ever read about LastPass (and people have focused its security a LOT) is that a bad guy can modify the JavaScript utilised by the extension if they took control of LastPass's servers, and have your plain text master password sent to a third party (assuming no cross-site protections).
The actual password database is fairly secure. As is the login process (which can further be strengthened with 2F and various options in the account settings).
I've had a _great_ experience using KeePassX on both Windows and Linux.
I use a sometimes-synced copy of the database on KeePassDroid on my Android phone. Actually, the user experience of KeePassDroid can only be described as vile, but that it works at all (allowing me to have all of my passwords securely available on my person) is awesome enough.
I prefer MacPass over KeePassX, because it support Keepass2 files by default - KeePassX only does in the non-stable version. That's the only annoyance with Keepass: it has two non-compatible file types and you need to stick with one (and hence the supporting software).
I don't particularly favor putting all my passwords in an online password manager. So I have some JavaFX gui I made that encrypts a password file (passphrase => PKDF => AES). In total it's like 200 lines of code - GUI, storing pass, generating pass, and rotating master password. The encrypted file I keep synced in my cloud storage.
But in retrospect I don't know if this makes any real difference from something like keepass. My encrypted file is transferred over some secure socket, so an attacker can at least a copy of the encrypted file if they either hack the cloud storage provider or somehow hijack my connection.
It's not exactly super portable but for sites I care about, I wouldn't log onto them on untrusted computers anyway.
I use KeyPass and sync it using BitTorrent sync on all my devices. The problem with my current setup is that I carry the keyfile along with the database which useless.
The only legitimate security gripe I've ever read about LastPass (and people have focused its security a LOT) is that a bad guy can modify the JavaScript utilised by the extension if they took control of LastPass's servers, and have your plain text master password sent to a third party (assuming no cross-site protections).
The actual password database is fairly secure. As is the login process (which can further be strengthened with 2F and various options in the account settings).