No. DHS will be able to communicate a traveler’s ESTA status to the carriers. However, DHS recommends that travelers print out the travel authorization application response in order to maintain a record of their traveler authorization application number and to have confirmation of their ESTA status.
U.S. regulation does NOT require a Visa Waiver Program (VWP) traveler to present a printed copy of the ESTA authorization page in order to travel. However, it is recommended that you print a copy for your records. If you did not receive or cannot locate your application number, you may retrieve the application through the ESTA website by entering your passport number, country of citizenship, passport issuance date, passport expiration date, and date of birth.