No. DHS will be able to communicate the traveller’s ESTA status to carriers.
According to the Department of Homeland Security, travellers are advised to make a screenshot of the response to the travel authorization request. This will allow him to keep the confirmation of ESTA status and also to keep a record of the travel request number.
U.S. law does not require a Visa Waiver Program (VWP) traveller to submit a printed copy of the ESTA travel authorization page. However, it is recommended to print a copy and keep it in your file.
If you have already applied for ESTA and have forgotten your application number, you can retrieve the application on the site by indicating the passport number, date of birth, name of the applicant and the country of issue of the passport.