The port is owned by the nation and the operation is entrusted to the Dover Harbour Board, created by statute and appointed by the DfT.
From the DHB website: "King James I gave Dover Harbour Board its Royal Charter in 1606 placing the Lord Warden and his "Board" in charge of running the port and since then the constitution of the Board has been changed on a number of occasions by Acts of Parliament to its present status of eight, a Chairman, and four other members appointed by the Secretary of State, two ex officio members and an eighth member appointed by the other seven. The present Board was created and incorporated by statute and is, therefore, a statutory body and has no shareholders to whom it is answerable."
Messrs Goldfield and Krayenbrink were asked about the wording of the Royal Charter at a recent presentation at the Town Hall. They replied that the charter is written in medieval Latin and that their mandate to run the port is a document of recent years, presumably the latest equivalent of the original charter.
This is a scan of the Royal Charter of James I, taken from "The Book of Dover" by the late Ivan Green. The caption reads: "The charter of James I transferring control of the harbour from the town to his newly-constituted body."
This is a scan of the top corner of the Charter, taken from the DHB publication "Dover Harbour - Royal Gateway" by Derek Leach. The caption reads: "The ornate and illuminated first letter of Jacobus (James) begins the 1606 Royal Charter."