If you don't have major scaling concerns in your future, it may not add any security to have a password, though please note that "not having a password" is not the same "trusting anyone who claims to be your application."
If your application runs as its own user on the server *, some databases may allow you to use 3rd-party authentication (e.g. what Postgres calls
peer authentication) which allows the server to use network or operating-system mechanisms for determining which users are who they say they are. For instance, under the right config, shell-user "bob" could access database-account "bob" without giving a password. Having things be simple can make securing your database and application easier.
That said, as jrtapsell's answer hints, you may need to move your application to another server for cost or performance reasons. In that case you will likely need a stored password or other secret like a private key, although there are some methods primarily within private networks which could be used to avoid this.
If you think there's a decent chance you'll need a password (or other stored-secret) in future, then you should just make the arrangements now so they are not made insecurely later by someone with less time or experience.
* Ideally, that account should be strongly passworded, or only available to log into via
sudo or equivalent mechanism