I have just copied this from BA's website:

What do I do once I’ve applied for an ESTA?

•You will be given a reference number confirming your application, which you should keep for your own reference.
•You will not be asked to provide this to us, it is automatically verified when you check in.


In light of the above, it is unreasonable to expect a passenger to keep a copy. The reference number may be useful, however, and I've just added it to my list of things to bring for the next trip.