The decision comes after the group posted footage online showing two people inside the base at RAF Brize Norton in Oxfordshire.
The clip shows one person riding an electric scooter up to an Airbus Voyager air-to-air refuelling tanker and appearing to spray paint into its jet engine.
The incident is being also investigated by counter-terror police.
The Home Secretary has the power to proscribe an organisation under the Terrorism Act of 2000 if she believes it is “concerned in terrorism”.
Proscription will require Ms Cooper to lay an order in Parliament, which must then be debated and approved by both MPs and peers.
