Nobody at the Ministry knew. They knew what the Dark Mark meant, of course, but they didn't know Voldemort was branding his inner circle with it. Nobody knew so nobody looked.
Snape strode forwards, past Dumbledore, pulling up the left sleeve of his robes as he went. He stuck out his forearm, and showed it to Fudge, who recoiled.
Fudge stepped back from Snape, too. He was shaking his head. He did not seem to have taken in a word Snape had said. He stared, apparently repelled, at the ugly mark on Snape's arm, then looked up at Dumbledore and whispered, 'I don't know what you and your staff are playing at Dumbledore, but I have heard enough. I have no more to add.'
Harry Potter and the Goblet of Fire - p.616 - Bloomsbury - Chapter 36, The Parting of the Ways
Now, I know Fudge is in full denialist mode right now, but the Minister of Magic himself does not seem to have any notion of what that brand truly indicated. There is no evidence that the matter of the brand ever came up during Death Eater trials and I conclude that nobody at the ministry was aware of Voldemort's practice of branding his Death Eaters.
Now of course if anybody had rolled up a prisoner's sleeve that might have created issues - why are you wearing Voldemort's mark as some kind of demented tattoo? - but the Death Eaters were smart enough to largely avoid capture prior to Voldemort's downfall - at which point the mark had faded anyway.
As for whether it was such a good idea in the event that a Death Eater got caught in service - well, I don't think Voldemort really wanted his Death Eaters denouncing him to stay out of prison, so I'm not really sure what you think he stands to lose by branding them. In fact, if anything, he seems to be trying to enforce loyalty. Just witness the scene in the graveyard:
'... we are still united under the Dark Mark, then! Or are we?'
'I smell guilt,' he said. 'There is a stench of guilt upon the air.'
'I see you all, whole and healthy, with your powers intact - such prompt appearances! - and I ask myself ... why did this band of wizards never come to the aid of their master, to whom they swore eternal loyalty?'
'And I answer myself,' whispered Voldemort, 'they must have believed me broken, they thought I was gone. They slipped back among my enemies, and they pleaded innocence, and ignorance, and bewitchment ...'
'It is a disappointment to me ... I confess myself disappointed ...'
'Stand up. You ask for forgiveness? I do not forgive. I do not forget. Thirteen long years ... I want thirteen years' repayment before I forgive you.'
'And yet you ran from my Mark, when a faithful Death Eater sent it into the sky last summer? ... Yes, I know all about that, Lucius ... you have disappointed me ... I expect more faithful service in future.'
'The Lestranges should stand here,' said Voldemort quietly. 'But they are entombed in Azkaban. They were faithful. They went to Azkaban rather than renounce me ... when Azkaban is broken open, the Lestranges will be honoured beyond their dreams.'
Harry Potter and the Goblet of Fire - pp.561-4 - Bloomsbury - Chapter 33, The Death Eaters
Voldemort is a psychopath, a megalomaniac, his Death Eaters are his, branded with his mark. When you enter his service you are his. Forever.