They didn’t - Fudge brought a Dementor and passed it off as accident.
When Fudge heard that they’d captured the one responsible for what happened at the Third Task, Fudge was insistent that he have a Dementor to guard him, supposedly to ensure his personal safety around a dangerous prisoner.
“When we told Mr Fudge that we had caught the Death Eater responsible for tonight’s events,’ said Snape, in a low voice, ‘he seemed to feel his personal safety was in question. He insisted on summoning a Dementor to accompany him into the castle. He brought it up to the office where Barty Crouch –”
- Harry Potter and the Goblet of Fire, Chapter 36 (The Parting of the Ways)
Once the Dementor got into the room with Barty Crouch Jr. it sucked out his soul, supposedly approved of by no one, acting on its own accord.
“The moment that – that thing entered the room,’ she screamed, pointing at Fudge, trembling all over, ‘it swooped down on Crouch and – and –’
Harry felt a chill in his stomach, as Professor McGonagall struggled to find words to describe what had happened. He did not need her to finish her sentence. He knew what the Dementor must have done. It had administered its fatal kiss to Barty Crouch. It had sucked his soul out through his mouth. He was worse than dead.”
- Harry Potter and the Goblet of Fire, Chapter 36 (The Parting of the Ways)
However, it may have actually been acting on orders. Dumbledore seemed very much to suspect Fudge had actually intended that to happen - Barty Crouch Jr. was dead, and he could continue on covering up the return of the Dark Lord.
“By all accounts, he is no loss!’ blustered Fudge. ‘It seems he has been responsible for several deaths!’
‘But he cannot now give testimony, Cornelius,’ said Dumbledore. He was staring hard at Fudge, as though seeing him plainly for the first time. ‘He cannot give evidence about why he killed those people.”
- Harry Potter and the Goblet of Fire, Chapter 36 (The Parting of the Ways)
The decision was never made by the Ministry as an organization, or officially, by Fudge either. However, it seems very likely that Fudge wanted the Dementor to suck out Crouch's soul and gave the order himself. However, Fudge had the excuse that the Dementor acted on its own, which he could have told the Ministry, and anyone else, should they question him.