If you find a third party company they could possibly replace the screen. That may allow the device to work so that you can enter a passcode & use iTunes to make a backup. It really depends on why the device does not work. A good third party repair company may also have other ideas to get the device working.
iTunes needs the screen to work to setup 'pairing' (you have to agree on the device on first connection to a computer) and to make a backup (the passcode is needed to allow a backup to begin).
If the device was setup to sync to iCloud & it has a working Wi-Fi you can backup to iCloud. iCloud backups happen when the device has Wi-Fi to the internet & also is on charge. You will need to free space to allow that to happen (or upgrade to more space).
The problem is that with out the screen you cannot tell if it is working correctly at all (besides checking for sound/ vibrations when attaching the power cable). Leave it on charge for a few days & view your iCloud account details to see if the device has a new backup - iCloud settings on a Mac shows the backups on iCloud if I recall correctly.