We present a novel technique for measuring the characteristics of a magneto-optical trap (MOT) for cold atoms by monitoring the spontaneous emission from trapped atoms coupled into the guided mode of a tapered optical nanofiber. We show that the nanofiber is highly sensitive to very small numbers of atoms close to its surface. The size and shape of the MOT, determined by translating the cold atom cloud across the tapered fiber, is in excellent agreement with measurements obtained using the conventional method of fluorescence imaging using a charge coupled device camera. The coupling of atomic fluorescence into the tapered fiber also allows us to monitor the loading and lifetime of the trap. The results are compared to those achieved by focusing the MOT fluorescence onto a photodiode and it was seen that the tapered fiber gives slightly longer loading and lifetime measurements due to the sensitivity of the fiber, even when very few atoms are present.