Two things:
Akeeba Backup 3.8.2 is not fully compatible with Joomla! 3.2. Please upgrade to Akeeba Backup 3.9.1.
The Download ID is entered in the component parameters page. This page is not handled by Akeeba Backup. It is handled by Joomla! itself, namely the com_config component shipped with Joomla!. There is a known issue in Joomla! 3.2 affecting a small number of servers that makes it impossible to save component options. You will observe that you have the same problem with the Options in other components, even those shipped with Joomla! e.g. Content, Media, User Manager and so on. Since the problem lies in Joomla!'s code we cannot provide a fix. The only thing we can tell you is to either wait for an updated Joomla! version or move your site to a different host.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!