Ummm...dunno. I suppose Keypass, although I've never used it. There are no doubt some other options too. I use LastPass because it's easily portable among Linux, Windows, Android, Apple, etc, and I have a multitude of computers/devices at work and home that I use. I can't carry around a thumb drive or authentication tool, so I'm kind of stuck.
Having said that, I'm certainly at the mercy of LastPass if it gets hacked in a way that breaks the encryption of the remotely stored password files. It seems unlikely (given this: https://www.lastpass.com/how-it-works), and I feel "reasonably" safe using it. In this case the benefits outweigh the risks (IMO). I've not implemented it yet, but it also supports Two Factor Authentication (including DUO, which I use for my Virginia Tech account), so that adds a bit more to my comfort level as well, and I'll no doubt implement it soon.