I've been working with the DS3231 a little bit, which is a newer version similar to the DS1307. I understand the internal registers are a bit different between the 2 versions. http://datasheets.maximintegrated.com/en/ds/DS3231.pdf
At least in the 3231, you can set an alarm which can generate an output signal which you can connect to an Arduino. If the Arduino has to act immediately when the time comes, then you could program an interrupt service routine in the sketch and use an interrupt input pin. Otherwise feed a regular digital input pin on the Arduino, and let the sketch loop, reading the input over and over until it changes.
Except for setting the time, alarm(s), and registers in the 1307/3231, you almost don't need the Arduino.