Dementors are under the control of the Ministry of Magic, at least for the duration of all seven Harry Potter books. During Harry's hearing for violating the Decree for the Restriction of Underage Wizardry in Order of the Phoenix, Dumbledore discusses this with Umbridge, and Fudge confirms Dementors are under Ministry control:
‘If it is true that the Dementors are taking orders only from the Ministry of Magic, and it is also true that two Dementors attacked Harry and his cousin a week ago, then it follows logically that somebody at the Ministry might have ordered the attacks,’ said Dumbledore politely. ‘Of course, these particular Dementors may have been outside Ministry control –’
‘There are no Dementors outside Ministry control!’ snapped Fudge, who had turned brick red.
Dumbledore inclined his head in a little bow.
‘Then undoubtedly the Ministry will be making a full inquiry into why two Dementors were so very far from Azkaban and why they attacked without authorisation.’
Order of the Phoenix - pages 134-135 - Bloomsbury - chapter eight, The Hearing
And Dumbledore reiterates this at the Sorting Feast once the students have arrived at Hogwarts for Harry's third year:
Dumbledore cleared his throat and continued. ‘As you will all be aware after their search of the Hogwarts Express, our school is presently playing host to some of the Dementors of Azkaban, who are here on Ministry of Magic business.’
Prisoner of Azkaban - page 72 - Bloomsbury - chapter five, The Dementor
It's explicitly stated in Prisoner of Azkaban that the Dementors are the guards of Azkaban -- they are not the administrators of the prison itself. Logic suggests that witches and wizards must work at Azkaban, too. And while it doesn't reference Azkaban per se, in Prisoner of Azkaban, in the chapter The Marauders Map, while in the Three Broomsticks, Fudge references a Ministry department called the "Magical Law Enforcement Patrol. Also in Prisoner of Azkaban, we learn that Fudge visited Azkaban himself, where he speaks with Sirius Black and gives Sirius his newspaper (which is how Sirius finds out Peter Pettigrew is stil alive -- he sees Peter, in the Animagus form of Ron Weasley's rat Scabbers, on the front page of the Daily Prophet):
‘Is it true he’s mad, Minister?’
‘I wish I could say that he was,’ said Fudge slowly. ‘I certainly believe his master’s defeat unhinged him for a while. The murder of Pettigrew and all those Muggles was the action of a cornered and desperate man – cruel ... pointless. Yet I met Black on my last
inspection of Azkaban. You know, most of the prisoners in there sit muttering to themselves in the dark, there’s no sense in them ... but I was shocked at how normal Black seemed. He spoke quite rationally to me. It was unnerving. You’d have thought he was merely bored – asked if I’d finished with my newspaper, cool as you please, said he missed doing the crossword. Yes, I was astounded at how little effect the Dementors seemed to be having
on him – and he was one of the most heavily guarded in the place, you know. Dementors outside his door, day and night.’
Prisoner of Azkaban - page 155 - Bloomsbury - chapter ten, The Marauders Map
The theme that I've come away with over the years, after numerous re-reads of canon, is that many innocent people are sentenced to Azkaban for very terrible crimes; therefore, the Ministry, going by the prisoner's crime, rather than guilt versus innocence, justifies the abominable treatment of all its prisoners. Fudge claims that wizards who commit minor crimes do not go to Azkaban:
‘Hang on,’ blurted Harry, ‘what about my punishment?’
‘I broke the law!’ Harry said. ‘The Decree for the Restriction of Underage Wizardry!’
‘Oh, my dear boy, we’re not going to punish you for a little thing like that!’ cried Fudge, waving his crumpet impatiently. ‘It was an accident! We don’t send people to Azkaban just for blowing up their aunts!’
But this didn’t tally at all with Harry’s past dealings with the Ministry of Magic.
‘Last year, I got an official warning just because a house-elf smashed a pudding in my uncle’s house!’ said Harry, frowning. ‘The Ministry of Magic said I’d be expelled from Hogwarts if there was any more magic there!’
Unless Harry’s eyes were deceiving him, Fudge was suddenly looking awkward.
Prisoner of Azkaban - page 38-39 - Bloomsbury - chapter three, The Knight Bus
True justice doesn't exist under Fudge's Ministry, and things only get worse under subsequent ministers, at least through the end of book seven. J.K. Rowling has said that Harry went on to work under the Ministry of Magic's Department of Magical Law Enforcement following Voldemort's defeat and death, with Kingsley Shaklebolt as Minister For Magic. She also specifically says that the Ministry ceases using Dementors as guards for Azkaban, for exactly the problems you cite in your question.