Problem with turning LCD off

PaulS:

If a port (6) on the Arduino is high

Which Arduino as 6 ports? Pins and ports are not the same thing, by a factor of 8.

  pinMode(power,INPUT); //sets 6 as input

But, without using the internal pullup resistor. So, you need an external pullup or pulldown resistor, or something supplying a distinct HIGH or LOW. What is wired to the PIN?

I'm sorry, I meant pin! I'll change that.
I am also using a pull down resistor connected to the switch.