If you no longer actively use an email account but want to retain it alongside your other email accounts, Postbox will keep attempting to connect to the server even if you deactivate downloading mail in the preferences. Unfortunately, in the current Postbox 3.0.11 there is no easy way to fix this.

This used to work, but this is no longer the case:
Step 1. Open the Options dialog and click on Advanced > Config Editor. Confirm.

Step 2. Filter for
server:server

Step 3. Make sure the list is sorted by the column Preference Name. Look in the column Value for the name of the email account you want deactivated in Postbox. The corresponding entry in the column Preference Name should look like this:
mail.server.serverXX.name
XX are digits that identify all server settings for this account. Make a note of this identifier.

Step 4. In the column Preference Name, look for the parameter:
mail.server.serverXX.defer_get_new_mail
and set it to
false
Postbox will still try to connect to the server, even though it clearly should not.

What you can do is recreate the folder structure in another functioning mail account of yours, then use Postbox’ Copy command to clone the messages from the non-functioning email account to the directory on your working mail account (for example in its Archive). Next, you need to remove the account that you can no longer access, from the user interface of Postbox so that it stop bugging you for login credentials.

In order to have an account disappear form the user interface, you can remove its identifier from the value of the preference setting:
mail.accountmanager.accounts

This way, you should be able to retain your mail in Postbox and not be bothered with password requests, in other words: eat your cake and have it, too.