When 3D printing a hollow box, what is the best course of action?
That depends very much on what your goal is with the box. If it needs to be hollow and you don't need any access to the inside (and also prefer it to be printed in one piece) than the answer provided by user77232 would probably be the best.
Alternatively, if you need access to the inside of the box you would probably be best off printing the box in two pieces. A box and a lid. This way you can print the box without support and also save on material that would otherwise be used as infill.
The amount of light coming into your box also depends on the translucency of your material and brightness of the light source. To find the required wall thickness would be a matter of testing with the desired material.