A true transparency, fading from the outside curve to the inside curve is quite tricky: it's relativity easy to get what looks like a transparency, (just fading into a uniform background fill).
Option1) keep it vector and use multiple objects - Create the moon shape. Create the inner 'sliver' shape. Blend between them. Change the inner's transparency to uniform, 100%. Select the blend. Break it, ungroup it, "Simplify" it.
The 'simplify' is a bit intensive, so be patient. The end result will be progressive shapes of more and more transparency; the more shapes (ie more steps in the blend) the smoother it will be. Change the acceleration of the blend to change the acceleration of the transparency.
(The banding is due to me removing the outline, but you could select the complete thing, convert it to bitmap, add some noise and blur it a bit to get rid of it)
Option2) Create a bitmap "shadow" and use this for a transparency mask - Crate a circle. Fill with black. Create a drop-shadow from it. Draw an enclosing box bigger than it. Fill box with white. Put box behind shadow. Select all. Export selected (BMP/GIF). Create moon shape. Use bitmap transparency and load image just saved. Stretch bitmap to suit.
I can't think on any other methods that would give as good results.