With products such as LastPass that store encrypted passwords on remote servers, what are the associated risks that aren't applicable to locally stored and encrypted passwords? If something is encrypted well enough, it's unlikely anyone would be able to break the encryption no matter how much processing power they have so I don't see any problems. Basically I'm asking why (even on this site) there is so much negativity and mistrust towards storing encrypted passwords over the internet?
If something is encrypted well enough, it's unlikely anyone would be able to break the encryption no matter how much processing power they have so I don't see any problems.
And that's a huge if! You have no control over how your data is encrypted with a service like LastPass. You can only trust that they have done so properly and that they do not have malicious intent.
Contrast this with an application like KeePass that stores your data locally. You control the application, you know how it's encrypted. You can take proper steps to secure the key file (if you are using one).
(Shameless advertising, I wrote a blog post about this about a year ago: http://www.infosecstudent.com/2013/02/switching-password-mangers-why-lastpass-just-isnt-working-out-anymore/)
The Internet isn't the security risk. Assuming all your stuff is locally encrypted and the password is reasonable and isn't known to the attacker, then you're golden.
Now, what might affect that? Choosing a bad password obviously, but more interesting is a local compromise on one of your workstations. If a piece of local malware gets the key to your lastpass or keepass or chrome or any other password storage database, then you lose. This is irrespective of whether that encrypted database is accessible over the Internet. The risk is your workstation.
So choose good passwords, don't re-use your passwords, but more than anything, don't use your password on a potentially-compromised workstation.