Almost all LCD displays have a data sheet on the 'net' if you Google it. That will give the current draw for the backlight Or you can use a voltmeter and measure the current. The Arduino suggests you stay within the 20ma range.
I measured several 4x20 LCDs for current draw and found they used a low of 53ma to a high of 275ma. I also have a 16x2 that uses 152ma. Trying to run any of those from the power of an Arduino pin would invoke the law of diminishing returns. As in one cooked Arduino chip.
A better/safer way would be to use one output pin to drive a transistor and have that act as a switch to turn the LCD backlight on and off. There are plenty of schematics for these simple switches around.
Mel