What are you willing to do to get to the data? Also, can you describe the damage in more detail? Is it just the touchscreen or do you suspect it's something more?
It is possible to get a phone repaired. If it was just the touchscreen, then Apple might be able to replace it to get it working, but they make zero assurances that it won't be wiped or perhaps they might give a replacement in lieu of a repair. But I find it tricky to recommend Apple service when the most important thing is the data, since they make it clear that they're never responsible if the data isn't recoverable or is wiped.
You might consider a third party repair shop that might be able to fix the screen or perhaps just use a working one to be able to "Trust" a computer in order to perform a backup. If the damage is more extensive, it might require board level repairs. This is something Apple doesn't specifically approve.