If you retrieve the booking using the booking reference and passenger's last name on ba.com, without logging into your EC account, you will see the itinerary. Below the summerised itinerary, there is a section: Additional services for your trip; there it gives you the option to add an EC number to the booking.

Several times, when making a booking over the phone, I give my EC number after giving my personal details, but the agent never puts it on the booking!