This is the way I do it, it only works if the texture below is all solid colour vector shapes, no raster images.
If your background texture has any shapes with strokes, you will first want to “object->expand” those shapes before doing this.
First make a simple square shape larger than your texture and crescent, choose a simple colour that doesn’t have to do with the design(ex: bright green/pink).
Make the layer order so that your crescent shape appears on top of the new square shape, now select both the crescent and square, open your pathfinder window and hit the “merge” button. Right click on this new merged shape and “ungroup”, select the crescent portion and delete it or hide it (hiding it is a good idea if you want to still utilize the shape afterwards). Now you have your “cookie cutter”, a solid colour square with your crescent hole cut into it. With the cookie cutter as the top most layer, select all the shapes of your background texture and the cookie cutter shape, again go into the pathfinder window, and again hit the “merge” button. After the merge, right click to “ungroup” this new shape and then select and delete your cookie cutter part. You should be left with your texture cut into the crescent shape, group everything together for easy positioning/transformation and you’re good to go.