A degree is great for some jobs, but for a nanny, I think experience coupled with some classes is ideal. For raising kids there is nothing like some hands-on experience, but I think the training part is very important. Everyone looking for a nanny position should at least take the time to get CPR certified. Doing some sort of enrichment activity (taking classes, reading child development books or magazines, browsing websites for learning activities, etc) shows that not only do you care about children, but that you are interested in being the best you can be.
I actually earned my BA as a Child Life Specialist. However, I have worked as a preschool teacher as my first job. I believe that experience is VERY important and that I wouldn’t be worth my wage without it. However, I also have to give TONS of credit to my educational background. It’s an extra bonus.
Having a degree proves to many employers that you have taken your love of children very seriously. It shows that you have put extra time and effort into learning about child developmental theories, etc.