The requirement of SSN to claim a child as a tax deduction went into effect during the time I had young children. My guess is @1987.
ETA: guess my memory is pretty good. http://money.usnews.com/money/blogs/planning-to-retire/2011/05/06/why-we-assign-social-security-numbers-at-birth
The Tax Reform Act of 1986 required that every dependent age five or older listed on a tax return have their own Social Security number, which led to a spike in demand for Social Security numbers for children at earlier ages. The Social Security Administration developed an enumeration-at-birth process in 1987, which quickly became the way the majority of people apply for Social Security numbers. Now parents indicate on the birth certificate form whether they want a Social Security number assigned to their newborn child. When the state vital statistics office receives the request with the birth registration data from the hospital, it forwards the information to the Social Security Administration and a number and card are issued for the child.