KeePass, or any other offline password manager, is a good first step. I really shouldn't need to go into the inherent issues with using an online password management system. However, to improve the security of the database, go with two-factor authentication by adding plugins such as OtpKeyProv and configuring KeePass to use it in conjunction with a Yubikey token.
(Disclaimer: I am not associated with either the OtpKeyProv developer or with Yubico. I use them as examples based on past successes.)