Back up Iphone with broken screen

Do you use iCloud automatic backups? If so, plug the device in to a power supply where it should have WiFi access and leave it for 24 hours so it can complete an incremental automatic daily backup to iCloud.

Otherwise, no, there is no way to manually execute an iCLoud nor an iTunes backup with a working display to input commands via the touch interface.

If the screen is not functional, no, there's not. If you haven't backed up before the phone was damaged, it's usually too late. And Apple will not guarantee you won't lose data when they replace the screen.

What do you need specifically? Is yours working?

I would note that if you're taking it to Apple or an AASP, their "repairs" for more severe issues is usually done as a complete replacement where they trade in the customer's device for a service replacement. Apple also makes no guarantees about the data and considers it the responsibility of the customer to have a working backup to restore data if they issue a replacement or if they wipe the data off a repaired device.

If you can't do a backup because of a malfunction, then you would need to get it repaired - possibly not by Apple. What's the specific problem?

If you've backed up to itunes or icloud on a regular basis, then do this How to back up your iPhone, iPad, and iPod touch - Apple Support

IF you have no backup, then it too late to try back up data.