You should treat them, i.e. spray them. cutting the bad affected leave is not going to solve the problem.

Spray them with what?

There are options:
=1= NEEM oil spray
=2= Immunox
=3= home made : ; milk and water mixture .

1 and 2 are available at Lowes, Home Depot ...
There could be other ... (maybe Daconile ?)

Will any of that kill the mildew and not hurt my plants?

This post was edited by sneed on Sat, Jul 13, 13 at 11:30

None of those will kill it, but they will help control it. You have to be very regular with your spraying for the remainder of the season. I prefer the milk method myself. Yes, you should remove the heavily affected leaves. I save my Neem for bugs because I'm mean that way >:)

I am also on Long Island and when the heat and humidity hit I got powdery mildew on all my squashes. Tried cutting off the leaves but that did not do it. I sprayed with Neem three times. Once, them two days later again and two days after that again. That stopped it. Make sure you do both sides of the leaves (top and bottom) and the stems also. Plus do both the new clean leaves and the ones already with the mildew. Now I just spray weekly to control/stop the mildew.