If I initialize a serial port on a Arduino, and connect all the correct wires to it but never actually use serial communication does the board loose power because of this, even though I haven't actually used the ports?
Yes. Not very much, though. This is because the internal uart circuitry is running regardless of whether there is actual data on the pins. From the datasheet (sec 31.8.3.) it looks like a UART probably consumes about 200uA (0.2mA) on a 5V 16MHz 328p. (Hmm. About double that on a m1280 or m2560.)
is the uart circuitry running even if the serial port isn't initialized?
and if the serial port is being used more often would it consume more power or just the 0.2mA?
is the uart circuitry running even if the serial port isn't initialized?
I don't know.
and if the serial port is being used more often would it consume more power or just the 0.2mA?
probably not. Depending on exactly how it is connected.
These questions are the sorts of things that may require actual measurements to resolve. Or they may be irrelevant: many things on an Arduino board consume more than the 0.5mA that a configured serial port might be using.
Have you read Nick Gammon's tutorial on power-saving? Gammon Forum : Electronics : Microprocessors : Power saving techniques for microprocessors
Thanks for your help!