For outbound authentication: store passwords outside of the code in a strongly-protected, encrypted configuration file or database that is protected from access by all outsiders, including other local users on the same system. Properly protect the key (CWE-320). If you cannot use encryption to protect the file, then make sure that the permissions are as restrictive as possible.
For inbound authentication: Rather than hard-code a default username and password for first time logins, utilize a "first login" mode that requires the user to enter a unique strong password.
Perform access control checks and limit which entities can access the feature that requires the hard-coded password. For example, a feature might only be enabled through the system console instead of through a network connection.
For inbound authentication: apply strong one-way hashes to your passwords and store those hashes in a configuration file or database with appropriate access control. That way, theft of the file/database still requires the attacker to try to crack the password. When receiving an incoming password during authentication, take the hash of the password and compare it to the hash that you have saved.
Use randomly assigned salts for each separate hash that you generate. This increases the amount of computation that an attacker needs to conduct a brute-force attack, possibly limiting the effectiveness of the rainbow table method.
For front-end to back-end connections: Three solutions are possible, although none are complete.