DS1307 power failure problem
Schematic is good. Run this checklist:


Check your power supply for clean wave, specially on connect and disconnect times. If it pikes for more than 1V add 4.7uF capacitor. If it do not fix the problem, change power supply and renew DS1307.
Check 3V battery current capability, no voltage only, DS1307 need 200uA min.
Check for 3V bat having unwanted path discharging it. Specially if you are using protoboard.
Check I2C commands/software.