There is no fruit that is only the base of a flower. Fruits develop from the ovary of a flower after pollination and fertilization occur. The ovary, which sits at the base of the flower, contains the ovules that become seeds.
However, some fruits, like cucumbers, have female flowers that have a tiny, immature fruit at the base of the flower. This fruit will develop into the cucumber we eat after the flower is pollinated. So, while the fruit is not only the base of the flower, it does start there.
