Resistor set up to drop to 12V for LEDs
If the LED strips work from 9 volts, just connect the two 9 volt batteries in parallel - this will give you 9 volts, with twice the usable running time.

If you connect the batteries in series, then use a resistor to drop the voltage to 12, you will be wasting a third of the battery power in the resistor.  

To determine the appropriate resistor, you will need to know what current the LED strips draw.  With that information, you can use Ohm's law ( V = I x R ) to determine the required resistance, and the power formula ( P = V x I ) to determine the power rating of the resistor. (I in both calculations is current, in Amps.)