OK now, let's first sort out the "
XY Problem".
Why do you imagine you want to "save power"?
If it is connected to a USB port, then the whole concept of "saving power" is simply nonsense. The only part of the device that requires significant power is the LED illumination - about 24 mA if this is the common sort of 1602 LCD with the value of R8 on the board being "101" - 100 Ohms. 24 mA is just within the capability of an Arduino pin to control (you will lose most of a Volt in the process) and is arguably a reasonable economy if you are using batteries.
The HD44780 chip itself consumes something like 600 µA so unless you have made the necessary board modifications to the as-yet-unknown sort of Arduino board and are using its sleep functionality, that current draw is
complete nonsense compared to the operating current of the Arduino. And you have to disable the USB interface and pilot LED in order to save current in sleep mode in any case.
However, apart from the chip, the 11k contrast ladder on the board (resistors R1 to R5) will also draw another 450 µA
and if you have not corrected the classic blunder in a long series of amateur "tutorials" and modules which show a 10k contrast potentiometer connected to VCC - 5 V - as well as ground with the wiper to pin 3, then you are wasting
another 500 µA. The contrast potentiometer should
not be connected to 5 V, that end should be left open or - if using a 10k pot - connected back to the ground, which will make the contrast setting
much easier and save that 500 µA. (The correct value of the contrast potentiometer would be either 1k or 2k.)
Right, so the LCD module
apart from the illumination, draws just over 1 mA. Do not attempt to remove its power and you do not have to worry about setting all the control pins LOW
before removing the power or re-initialising it when you power it up again.