You can try changing the blending mode of the drop shadow. This will depend on the effect you want, what the underlying object is etc. Different blending modes will have different effects so experiment and see what works. Keep in mind that blending modes work differently in CMYK vs RGB so that may have an affect.
You can see in this example that setting the drop shadow to "Color Burn" only shows the drop shadow on the object below:
Use a clipping mask. Copy all the objects you want the drop shadow to cover then use pathfinder to create one shape from the objects and use that as a clipping mask.
This isn't ideal though as you need to edit the clipping mask as well as the contained objects if you make changes.