How can I crop everything outside of an art board in Illustrator?

I would make a rectangle the same size as the artboard and turn it into a clipping mask.

To do this, group together all the layers you wish to trim by selecting them and entering command+g. Next, make a rectangle with the same dimensions as your artboard, and center horizontally and vertically. With the rectangle layer in front, select both objects, enter command+7, or go to object>clipping mask>make, and voila. You will only see the parts within the dimensions of the rectangle. Simply double-click the new object to access the clipped information or enter command+alt+7 to get rid of the clipping mask.