I'm moving soon and will be towing my car (2003 Honda Civic EX w/ automatic transmission) about 1,600 miles. The car will be towed behind a box truck on a trailer, i.e. the wheels will not be on the ground. Do I need to do anything special to prepare the car for the trip? Like make sure my gas tank is full/empty?
Since the car will be on a trailer there's not much that you need to do.
Check the tire pressure, and fix any tires that have a leak. If the car is strapped down, depending on how you strap it, a tire loosing air could make the strap loose. This isn't likely to be catastrophic, but it's something that will be nice to avoid.
If you don't live in an area with a lot of rain, you might want to make sure all the windows are tight and leak free by running through a car wash a few times. In 1600 miles you're likely to see some rain. Close the sun roof.
Other than that, make sure the car is in good running condition so it's not a stranded lump on the trailer when you get to the destination, and remember to lock the car while it's on the trailer just like you would anywhere else.