Without code it is difficult to isolate the issue, but if there is any doubt you may use the read and write routines from here: I2C Clock The time and date registers are the same addresses for 1337 and 1307 and that code I know is without problem.

Beyond, I would verify resistors on I2C lines, "$" to designate hex values and, just because it caused me a problem, try a new crystal (and verify capacitance requirements)? Some of this is unlikely by your symptoms listed, but sometimes just looking again...

Can you read the status register before applying values? Are "default" register values returning logical results? Any output from heartbeat (I think 1Hz at startup)? I read it requires a transistor to run an LED, but have not tested anything with it. Clock should run from startup, yes? Try only reads to eliminate "write" problems and anything else you can think of to be confident it is "on" and ticking...

Can you verify data to clock chip on SDA line?