It all comes down to what the Ministry of Magic has to offer the Dementors
The point isn't about controlling the Dementors; it's about what the MoM has to offer. The Dementors are intelligent and sentient (semi-sentient at the least) enough. They obeyed the Ministry for years because, by guarding Azkaban, they were provided with sustenance of any remaining hope or happiness in the prisoners.
As seen in Harry Potter and the Half Blood Prince, the discussion between Fudge and the new British Prime Minister,
"I thought dementors guard the prisoners in Azkaban," he said cautiously.
"They did," said Fudge wearily. "But not anymore. They've deserted the prison and joined He-Who-Must-Not-Be-Named. I won't pretend that wasn't a blow."
Harry Potter and the Half Blood Prince (The other minister)
So the control is more of symbiotic than allegiance or obedience. The MoM provides them prisoners to feed upon, and in return, the dementors guard them from escaping.
This is also supported by what Dumbledore tells Harry later in the book: Voldemort's offer to the Dementors of free reins is what motivated them to follow him and abandon Azkaban.
Although it is not mentioned how the MoM controls the Dementor breeding, they do breed. As Fudge tells the Prime Minister in Harry Potter and the Half Blood Prince:
"But," said the Prime Minister, with a sense of dawning horror, "didn't you tell me they're the creatures that drain hope and happiness out of people?"
"That's right. And they're breeding. That's what's causing all this mist."
Harry Potter and the Half Blood Prince (The other minister)
As to why they didn't breed when they were at Azkaban, is not entierly known. One possible reason could be, they didn't need to. They had all the things they needed: prisoners to feed upon, and sometimes even souls to suck out (by Dementor's kiss). And as JKR mentions,
You cannot destroy Dementors, though you can limit their numbers if you eradicate the conditions in which they multiply, that is, despair and degradation.
So it is possible that by controlling their exposure to despair and degradation, the MoM controlled the breeding of Dementors.
I also add that all the ministry officials were well-versed with the Patronus charms. In Harry Potter and the Deathly Hallows, while Umbridge is interrogating Mrs. Cattermole, she is, at the same time patrolled by her patronus (while she is paying very little attention to it
The moment he had passed the place where the Patronus cat patrolled, he felt the change in temperature: It was warm and comfortable here. The Patronus, he was sure, was Umbridge’s, and it glowed brightly because she was so happy here, in her element, upholding the twisted laws she had helped to write.
Harry Potter and the Deathly Hallows (The Muggle-born registration commision)
As seen in other instances, even wizards like Lupin had to concentrate on their patronuses in order to be protected. Therefore, it is possible that the high-ranking MoM officials were well-versed with the Patronus charm.
In conclusion, by offering the Dementors prisoners to torment and controlling their [Dementor's] access to breeding conditions, the MoM was able to, in a way, control these foul creatures.
