The answer to this can be found right at the beginning of Part 3, Chapter 1. The answer is that Winston's serial number is "6079 Smith W."
What is interesting about this is that this is not a number that is given to him when he gets captured. Instead, it is a number that has been his throughout the book. We see it used for the first time in Part 1, Chapter 3. There, the fitness instructor on the telescreen uses it. This is a way of showing us how the whole society is like one giant prison. People are so unfree in everyday life that they don't even need a new number when they are literally imprisoned.