MB here. My nanny got her job in large part because we were impressed that she emailed a professional resume, cover letter, photo, and several reference letters shortly after we initiated email contact. All the letters were raves and gave us a real sense of her strengths and personality. Even aside from the content of the reference letters, her initiative in providing them before we'd even met showed a level of professionalism, organization, and attention to detail that other candidates were lacking. Of course we followed up with phone calls to the letter writers and a couple of other references before making an offer, but the letters definitely moved her to the top of our "wanted" list from the get-go.
Nannies need to remember that most people who hire nannies work in some sort of office environment. We're used to reviewing information, including employee backgrounds and credentials, on paper. We like seeing a person's history on paper because this makes it easier and quicker for us to glean relevant information. Providing reference letters is one way for a nanny to play to her prospective employers' comfort zone in this regard.
I've heard some nannies on this board talk about requesting reference letters throughout the term of employment, as often as every three months. As an MB, I would not look favorably upon such a request and would probably decline to comply. It seems overly paranoid to me, an indication that the nanny might be planning an unprofessional exit without proper notice, and an unwarranted drain on my time.
That said, I'd be happy to provide a nanny with a written performance review and have done so in the past. Of course, a performance review tends to be part of a prospective planning process, so it usually identifies goals for professional development or enhanced job performance (usually derived from changes in my children's developmental needs) as well as areas in which I'm thrilled with the nanny's performance.