Well if you're that worried about Dropbox going away and needing access to your passwords in between the point it shuts down and you noticing and setting up a replacement, use webdav with your own server, or hack up an rsync/cron-based concoction that will do roughly what Dropbox does in this context. There are 100's of ways to synchronize a file across computers, it's just that Dropbox is by far the most conveniet at this point in time.
I agree that this is probably a bit to complicated for normal users.
My personal problem with lastpass is that by default your passwords are recoverable which means that by default lastpass or someone with access to their system has access to your passwords (you can disable this and read the source to their obfuscated javascript app for chrome to make sure that it is really doing what they say, encrypting locally then sending).[1]
Keepass at least is opensource and it works well when you use dropbox so long as you aren't accessing it on many different computers on a daily/weekly basis, then it just becomes a pain in the ass.
[1] This may have changed but I don't think they would appeal to many users if it has, and they do have an option to disable it but the obfuscated javascript is what stopped me from looking further.
This causes the same problem as LastPass - reliance on a 3rd party service.