Why does the chance of having a baby with Down's syndrome increase with the age of the mother?
The frequency rises with maternal age due to a peculiarity of meoisis
  in female mammals.  Meiosis is originated in the fetal ovary,
  arresting at metaphase I with the homologous  chromosomes aligned for
  segregation. Cells remain in this state until the time of ovulation,
  often  decades later in humans. The longer cells remain in the
  arrested state, the greater the chance  that there will be a
  nondisjunction event when meiosis resumes.


Source : http://www.madsci.org/posts/archives/2005-07/1121714807.Ge.r.html

I think you have the misconception that egg cells have completed meiosis by the time female is born. This is not true ( as indicated above in the answer). At birth, they are in metaphase I , after puberty, each month some of them divide further but only one survives and get arrested at metaphase II . Only after fertilization is the meiosis complete.