Why does my toilet make a dripping sound at the base where it attaches at the floor for a short time and stops when I either flush or add water to the toilets base

It may sounds like the drip is at the base but I am betting you can't see anything, right? The sides of the toilet are hollow and water fills the rim and then flows down the sides of the inside bowl. The dripping could be from the rim. Try watching/listening with the lid raised.

How long does the dripping sound last? If it last for just a few minutes I would say you have a horizontal section of drain pipe that is not properly sloped (1/4)" per foot). A small amount of water in the horizontal pipe slowly falls down a vertical pipe in drops, creating the sound. If the sound never stops, I would say you have a very, very slow leak in the flush valve. The leaking flush valve slowly adds water to the bowl. This causes a very small amount of water to flow out of the bowl which falls down a vertical pipe in drops.