How did Barty Crouch Jr. and the other Death Eaters not get caught by the Ministry, when they have tattoos on their arms of the Dark Mark? Is the use of a cloak alone enough, no raids on their body parts? Even Barty Crouch Jr's father was shocked when he was revealed to be a Death Eater.
The Ministry didn’t seem to know about the Dark Marks.
When trying to prove to Fudge that the Dark Lord had indeed returned, Snape showed him his Dark Mark - and also explained what it was and what it meant. For Snape to think this explanation was necessary, it implies that Fudge, the Minister of Magic, would not know about them without having it explained first.
“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.
‘There,’ said Snape harshly. ‘There. The Dark Mark. It is not as clear as it was, an hour or so ago, when it burnt black, but you can still see it. Every Death Eater had the sign burnt into him by the Dark Lord. It was a means of distinguishing each other, and his means of summoning us to him. When he touched the Mark of any Death Eater, we were to Disapparate, and Apparate, instantly, at his side. This Mark has been growing clearer all year. Karkaroff’s, too. Why do you think Karkaroff fled tonight? We both felt the Mark burn. We both knew he had returned. Karkaroff fears the Dark Lord’s vengeance. He betrayed too many of his fellow Death Eaters to be sure of a welcome back into the fold.’
Fudge stepped back from Snape, too. He was shaking his head. He did not seem to have taken in a word Snape had said.”
- Harry Potter and the Goblet of Fire, Chapter 36 (The Parting of the Ways)
Further proof that the Ministry was unaware of the Death Eaters’ Dark Marks is that they were never used as evidence in the trials of suspected Death Eaters. We see some of these trials, and the presence of the Dark Mark on a suspect is never checked for or used as evidence. This likely means the Ministry didn’t know, since it would be useful evidence if they did.
It’s also likely the Death Eaters themselves did attempt to keep their being given Dark Marks secret from all but their fellow Death Eaters,and certainly from the Ministry. The Death Eaters wouldn’t have gone around showing off their forearm to the Ministry, and without the prior knowledge that the Dark Lord’s inner circle of Death Eaters had them, the Ministry wouldn’t likely think to just check the suspect’s forearms randomly. It’s also unlikely they’ notice the Dark Mark - most robes cover the forearms, and by the time most Death Eaters were on trial, the Dark Lord had fallen, so the Dark Mark would have faded.
The mark made it easier to recognize someone as a Death Eater, but it faded as Voldemort's health faded.
An inactive Dark Mark appears as a faint mark on the inner part of the left forearm, similar to that of "a vivid red tattoo"; an active one is jet black.
The deepness of the Mark's colour is connected to Voldemort's physical health. When Voldemort suffered his first defeat, the Marks on the Death Eaters faded, but as he began regaining his strength, the faded marks became clearer
Also, not all Death Eaters got a Dark Mark, like Fenrir for example.
"The werewolf might be allowed to wear Death Eater robes when they wanted to use him, but only Voldemort's inner circle were branded with the Dark Mark: Greyback had not been granted this highest honour."
- Harry Potter regarding Fenrir Greyback's lack of a Dark Mark
In addition to Bellatrix's evidence, further evidence that the Dark Mark tattoos were not known about comes from Harry's conversation with Sirius in the cave in Goblet of Fire (my emphasis):
“Yeah, you should’ve seen Snape’s face when Karkaroff turned up in Potions yesterday!” said Harry quickly. “Karkaroff wanted to talk to Snape, he says Snape’s been avoiding him. Karkaroff looked really worried. He showed Snape something on his arm, but I couldn’t see what it was.”
“He showed Snape something on his arm?” said Sirius, looking frankly bewildered. He ran his fingers distractedly through his filthy hair, then shrugged again. “Well, I’ve no idea what that’s about ... but if Karkaroff’s genuinely worried, and he’s going to Snape for answers ...”
Karkaroff was clearly showing Snape his Dark Mark. Sirius knows that Karkaroff was a Death Eater. Sirius had also been in the original Order of the Phoenix. If anyone would have known about the Dark Mark tattoos it would have been the Order. Yet when Sirius hears that a known Death Eater shows something on his arm to a suspected Death Eater, he has no idea what that's about. It seems pretty clear that Sirius was simply unaware of the tattoos.
However, it seems that even before the second war began, the fact that Death Eaters have the mark was already better known. For one, Snape must have told Dumbledore about it at some point because we see in his memories in Deathly Hallows that in the months leading up to Voldemort's return he told Dumbledore that it was getting darker:
“Karkaroff’s Mark is becoming darker too. He is panicking, he fears retribution; you know how much help he gave the Ministry after the Dark Lord fell.”
Additionally, Barty Crouch Jr. clearly references the Dark Mark to Snape in Goblet of Fire while masquerading as Moody, so presumably he thought it would be in character for Moody to know about the Dark Mark:
“ ’Course Dumbledore trusts you,” growled Moody. “He’s a trusting man, isn’t he? Believes in second chances. But me — I say there are spots that don’t come off, Snape. Spots that never come off, d’you know what I mean?”
Snape suddenly did something very strange. He seized his left forearm convulsively with his right hand, as though something on it had hurt him.
Most importantly, perhaps, by the time we get to Order of the Phoenix Hermione is already assuming that lack of a Dark Mark would be sufficient evidence to show that one is not a Death Eater (my emphasis):
“The trouble is,” she said to Harry, “until V-Voldemort — oh for heaven’s sake, Ron — comes out into the open, Sirius is going to have to stay hidden, isn’t he? I mean, the stupid Ministry isn’t going to realize Sirius is innocent until they accept that Dumbledore’s been telling the truth about him all along. And once the fools start catching real Death Eaters again it’ll be obvious Sirius isn’t one ... I mean, he hasn’t got the Mark, for one thing.”
The fact that the Ministry doesn't know about the Dark Mark does not make this problem go away. Why would Karkaroff not use this knowledge to get a better deal to get out from Azkaban? When he says only Voldemort knew all his Death Eaters, it would make sense for him to reveal the fainted Dark Mark, and he would tell the Ministry that the Death Eaters are likely to have it, and it would be quite weird for someone else to be wearing the mark considering what it means. Snape really should have told Dumbledore and either of them would most likely tell the Ministry as well. The only way it would have worked, was if the Dark Mark disappeared entirely.