Eve Tushnet found the Time Magazine Cover archives - her birthday cover was "Search for a Pope" (1978!). So I looked up mine and it was "The Catholic Exodus: Why Priests and Nuns are Quitting". (My exact birthday is Feb. 28th-for those thinking of gifts I like solid gold bars.)
The cover features James Shannon and Anita Caspary. James Shannon was apparently a bishop who left the priesthood and married. Ralph McInerny just wrote about him in Crisis Magazine. Anita Caspary was Mother General of the Immaculate Heart Sisters from 1963 to 1969 when a huge group of nuns left the Catholic Church. So the Time cover on the day I was born was about the confusing upheaval in the Church and Eve's was about the search for a Pope who would become JPII-interesting.