some days back my dad's phone fell down and was completely shattered.
the account sync was turned off and i couldn't do anything to backup the contacts because the touch wasn't responding at all ( display was visible )
so found an interesting way on itechify.com. Method 1
for this to work your phone should support OTG (USB on the go ) i guess most of the smartphones these days comes with one.
USB OTG Cable
USB Mouse ( i used a wireless bluetooth mouse , but any wired USB mouse will work )
USB data transfer cable.
just connect the OTG cable to the phone and the USB mouse to the OTG cable and voila , you will see a cursor on your phone's screen and you can go through menu's , do whatever you want Method 2
i haven't tried this but for several others this method work.
download a software called "my phone explorer" just google it and follow the steps.