We have just installed a new granite vanity top with side and back splashes. Do we need to caulk the base of these splashes where it meets the top?? If so, is there a special caulk that we need to use??

Yes you need to seal this area, otherwise you will find that water will get behind the wall in that area and cause issues. The installer should have done this as part of the install process. Typically they place a bead of clear silicon along the back edge of the top before they set the back splash stone. This is then wiped clean along the surface to form a clean bead. If they did not do this then you need to surface caulk along the edge with clear or color match caulk. Stay on this as it will break down and need to be done once and a while as cleaning the surface will cause the caulk to loosen and come off.
The type of silicon you need often can be found at a granite stone supplier, or tile store. The big box stores do carry the material that will work, but your options of color match other then clear, bone, white is limited. There are caulks that are mold resistant and usually have Microban lable on them. They often say on the tube kitchen/bath.