iTunes automatic downloads log-in prompt
Essentially, there are two different options you can try - depending on the way you want to use iTunes. I'll be writing this answer for Mac users, but Windows users can probably suss it out using these instructions.

Both options will only require you to click "Store" from the menu bar from within iTunes.



Option One (red) - If you intend on using the Store, you can simply click the "Authorize This Computer..." menu option. This will prompt you for your password and tie your computer and iTunes library to your iTunes account.

Option Two (blue) - If you don't intend on using the Store and would simply like to not see the dialog box prompting for your password, you can click the "Sign Out" menu option.

One of these two things should solve the problem of the persistent password prompt.

Best of luck!