Let's look at the risks.
Encrypted password database
To steal your passwords, the attacker has to steal the database. Then she has two options: steal the master password or guess/bruteforce it. Once the attacker has the database and master password, she has access to your current accounts until you change stored passwords.
Password generator (from a single master password)
To steal your password, the attacker has to steal your master password, or steal one of the derived passwords and guess/bruteforce the master password. Once she has it, she can generate any current and future passwords, until you stop using the generator with the same master password. Thus, using the password generator scheme (with at least one compromised derived password or the ability to quickly check derived password guesses) is roughly equivalent to the encrypted database scheme where the attacker has permanent read-only access to the encrypted database.
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There are other problems with the second scheme: most password generators I've seen are insecure: they use fast hash/HMAC to generate passwords. Thus, if your master password has low entropy, it's easy to bruteforce it.
The most sensible password generator I've seen is described in this paper: A Convenient Method for Securely Managing Passwords
. It uses a very slow hash to generate a secret seed from master password and stores it. It then hashes the master password again with a faster hash function, combines this hash with the stored seed and derives passwords from the result. This considerably slows down bruteforce attacks.
However, it's less safe in another way since you can never change an individual site's password if it is ever compromised.
It's possible to change passwords generated by the second scheme: just add some counter to the domain (e.g. "example.com" -> "example.com:2"), though this isn't convenient, as you have to remember the counter.
In both cases it seems that knowing the master password either gives you the other passwords, or allows you to generate them. Are their any other options? Am I missing any other large security problems these password managers create?
Unless you can remember many random passwords, using open source offline password manager with encrypted database (with correctly implemented encryption and key derivation) is probably the most secure scheme available. However, there are factors other than security to consider, for example, risk of losing the database, convenience, etc.
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The linked password generator is insecure (even if used offline) if the master password has low entropy: it uses a single round of HMAC-SHA256 to derive passwords.
