How to Power a Controller and Motor Driver off same 24V source
Powering the 'duino off of a single 12V battery, with other parts powered from the 2 batteries wired in series, would cause an imbalance of charge between the batteries. Whether or not this imbalance would be "enough to worry about" could be debatable in this instance (due to the difference in total consumption between the arduino &amp; the motors).
Luckily, if peak efficiency is the goal, #1 isn't something that needs considered.
While a resistive voltage regulator would run more efficiently from a 12V supply, a dc-dc converter, or switched-mode power supply (SMPS) can run just as efficiently from the 24V "system power" for the motors.
In this instance, I would recommend looking into a single-inductor [buck converter] topology.
Here's one I found in a quick Google search, for under $2USD, that's rated for up to 3A output current, with an input voltage range up to 28V.