If you have an iPhone with a broken screen it can sometimes be very difficult to get into the phone. I came across an issue where the screen was too badly damaged to be able to see the prompts, the phone was not connected to a network and it did not have a SIM as the owner had already purchased a new phone. However, the user wanted some information from the phone as it had not been backed up to a computer for a long time and iCloud back-ups had not been used.
The message from this is that always make sure you have a recent backup and that iCloud syncing is enabled for everything you need on the phone. I realise that storage beyond the 5 GB costs money but it is well worth it, especially if you have an issue such as this. This will eliminate the story you hear every now and then where someone says they lost all their contacts, appointments, messages etc.
The article covers some areas but may not work for everyone (it didn't for me and in my case suggesting to the user to get a screen replacement was probably the only real option).