Based on the answer Why does the Department of Mysteries keep prophecies? and the idea that prophecies can be studied, how could they study the prophecies if they can only be retrieved by those whom they relate?
The Keeper of the Hall of Prophecy can presumably raise and lower the defensive spells.
There is one person who's in charge of the Hall of Prophecy. (They may have a team of helpers but this isn't specified).
“The official record was relabeled after Voldemort’s attack on you as a child,” said Dumbledore. “It seemed plain to the keeper of the Hall of Prophecy that Voldemort could only have tried to kill you because he knew you to be the one to whom Sibyll was referring.”
(Order of the Phoenix, Chapter 37, The Lost Prophecy).
There's no further canon info on this role. But it seems pretty clear that the Department of Mysteries is divided into several different sub-departments working in different rooms (the love room, the death room etc.). The defensive spells which apply to the prophecies affect everyone apart from the people they were made about. This includes Unspeakables who work in the Department of Mysteries; Bode was struck down by the defensive charms, for instance.
“Why?” Malfoy sounded incredulously delighted. “Because the only people who are permitted to retrieve a prophecy from the Department of Mysteries, Potter, are those about whom it was made, as the Dark Lord discovered when he attempted to use others to steal it for him.”
(Order of the Phoenix, Chapter 35, Beyond the Veil).
So the defences hold for random trespassers into the Ministry and for Unspeakables who work in other areas of the Department of Mysteries. Whilst there isn't definitive canon proof that the keeper of the Hall of Prophecy can access the prophecies it'd be pretty hard for them to do their job if they couldn't.
Perhaps once a prophecy has come true or been circumvented and thus made null, it's no longer considered a 'prophecy' since the events have come true/been avoided, so the defensive spells aren't required anymore as, since the events have come to pass, nobody can use the prophecy to change them. This means that the Ministry could have someone examine the nullified prophecy without risk of disrupting future events and thus learn more about how prophecies work.