1. Possibly you have a pretty old version of plugin installed on cPanel server. It does not include functionality of automatic adding FTP accounts if needed for publication. Re-install plugin following installation instruction in your license (you need to uninstall and install it again). This should fix the problem.
2. Problem occurs when site builder cannot create FTP account when opening builder from cPanel. To fix this problem you need to ensure that you have not exceeded limit of FTP accounts in your cPanel account. Delete unused FTP accounts. There may be old FTP accounts left which were created by builder and not used anymore (they are of type "firstname.lastname@example.org"). As soon as you have space for at least one FTP account the website publication should work.
3. Make sure that "FTP Manager" is enabled on WHM system for your customers. Without it publication may not always work.
cPanel installation uses custom style for "paper_lantern" theme (this can be verified by lookin to folder /var/cpanel/customizations/styled/ what will contain folders with style name). To solve this issue call this command: "/usr/local/cpanel/bin/sprite_generator --theme paper_lantern --style [style_name] --verbose" as root in server cPanel is installed ("[style_name]" in this command needs to be replaced with a name of a style folder you want to fix).
cPanel blocs connections coming from builder for incompatibility related issues. Try to enable "Broken Clients Compatibility" option in FTP configuration within WHM system. Please check if firewall doesn't block passive FTP ports when using together with SSL.
Method 1 (if error appears when opening builder from WHMCS or WHM being logged in as admin):
Take the following steps to fix this problem:
1. Login to cPanel server per SSH;
2. Delete file /home/[user]/.spbldr_localStorage (this file is created by builder plugin);
Note: If you want to delete this file though cPanel file manager initially you may not see files starting with "." by default. To fix this click button "Settings" in top right corner of file manager page and check option "Show hidden files (dotfiles)".
3. Delete FTP accounts of type "email@example.com" (they were created by builder plugin);
4. Open builder from WHMCS (or WHM) again and publish website to check if this solved the problem.
Method 2 (if error appears when opening builder from customer cPanel account being logged in with customer's username and password):
1. Login to WHM;
2. Find customer with problem on "List accounts" page;
3. Change customer's password (you can set the same password as it was earlier);
4. Open builder from customer's cPanel account again and publish website to check if this solved the problem.
The reason of this error is that your builder domain is not working. Once you configure DNS for builder domain and domain starts working this error should go away.
Even if DNS is already propagated and builder domain is working it may not be resolved on your cPanel server due to some reason (for example server has not gotten new DNS configuration yet).
Builder requires FTP passive mode enabled on your server to be able to publish websites. Please refer this topic to enable FTP passive mode on cPanel: