Dumbledore is described as the only one whom Voldemort feared. Did he kill any Death Eaters, or anyone else for that matter?
We don't know for sure.
There's never any confirmed canonical instances of Dumbledore killing anyone, nor any confirmation in canon that he never did. However, there are various pieces of evidence supporting each side.
Evidence for YES
The following exchange between Dumbledore and Harry (emphasis mine) suggests that Dumbledore might have killed, presumably Death Eaters, in the past.
"Of course you were," said Harry. "Of course - how can you ask that? You never killed if you could avoid it!"
"True, true," said Dumbledore, and he was like a child seeking reassurance.
-- HP and the Deathly Hallows, Chapter 35: "King's Cross"
See also TheDarkLord's nice and exhaustive answer to the related question According to canon, did Dumbledore ever use an Unforgivable curse? (where I came across this quote).
When Snape comes to Dumbledore in the meeting that leads to him changing sides, he begs Dumbledore "don't kill me". See also Why would Dumbledore have killed Snape, and where was the light coming from? ... however, this doesn't necessarily reflect Dumbledore's natural reaction more than what Death Eater Snape thought he might do. As I said over there:
If the situations had been reversed - if Voldemort had met a lower-ranking member of the Order of the Phoenix - the outcome would most probably have involved the words "Avada" and "Kedavra" coming from Voldemort's lips. Of course Dumbledore would have been less quick to jump to the killing option, but it's questionable whether Snape would have understood this: hanging out with Death Eaters for so long, he might have forgotten how decent people behave.
Evidence for NO
Dumbledore didn't kill Grindelwald. Regardless of his personal feelings towards his enemy which might have prevented him from wanting to kill him, the fact remains that he was able to disarm and defeat this incredibly powerful wizard without killing him. If he could do that to Grindelwald, he could probably also do it to all the Death Eaters he encountered. Perhaps he never needed to kill anyone, but could instead defeat them without taking their lives. And as Harry said in the quote above, he wouldn't have killed without need.
In the few cases we know about in detail where Dumbledore has gone toe-to-toe with Death Eaters or even Voldemort himself (in particular, the battle in the Department of Mysteries), there were no confirmed deaths of bad guys. Dumbledore and the Order were able to defeat quite a number of Death Eaters without killing them.
It doesn’t seem likely he ever killed, not after Ariana’s death.
After Dumbledore thought he killed his sister, he tried very hard to stay on the “right” side of things, choosing to teach instead, and rarely involved himself in actual combat. He worked against the Dark Lord, but not directly in battle normally.
When we do see him in combat, he never seems to aim to kill or seriously injure. He didn’t kill Grindelwald, of course, but that in itself doesn’t prove he’d never kill anyone - Grindelwald was his old friend (and crush according to a JKR interview) so he might feel more sympathetic towards him. However, at the Department of Mysteries, he doesn’t try to kill Bellatrix, he traps her under a statue instead, and he doesn’t seriously injure her either.
“Voldemort raised his wand and another jet of green light streaked at Dumbledore, who turned and was gone in a whirling of his cloak. Next second, he had reappeared behind Voldemort and waved his wand towards the remnants of the fountain. The other statues sprang to life. The statue of the witch ran at Bellatrix, who screamed and sent spells streaming uselessly off its chest, before it dived at her, pinning her to the floor.”
- Harry Potter and the Order of the Phoenix, Chapter 36 (The Only One He Ever Feared)
She was certainly a dangerous Death Eater, and Dumbledore didn’t have any special attachment to her like he did with Grindelwald, so if he was willing to kill Death Eaters, she’d be a likely candidate. She was both powerful and extremely loyal - she’d be one of the Death Eaters who’s most lethal and least likely to change sides. He could have done more harm or killed her if he’d wanted to, but he didn’t. (He also doesn’t try to kill the Dark Lord but that’s most likely because he knew of the Horcruxes).
Since he didn’t kill her, I highly doubt he’d be willing to kill other Death Eaters, who were most likely less powerful than her and easier to take down nonlethally. It’s hard to imagine after him not killing her or really hurting her that any situation where he’d be willing to kill.
He may have accidentally killed Ariana, but it’s unclear.
Dumbledore blamed himself for his sister’s death, but it’s unclear who actually cast the spell that did it (and Fantastic Beasts may reveal a different reason for her death). He certainly blamed himself for it, though. He considered himself to blame for the circumstances of her death, and didn’t know who killed her but was terrified he might find out that it was him. Pottermore suggests it was likely Grindelwald, though - absolving Dumbledore from being the one who actually killed her.
The only thing that somewhat implies Dumbledore might have killed at some point seems to be more a matter of phrasing than an actual implication that Dumbledore killed intentionally at any point.
“Was I better, ultimately, than Voldemort?’
‘Of course you were,’ said Harry. ‘Of course – how can you ask that? You never killed if you could avoid it!’
‘True, true,’ said Dumbledore, and he was like a child seeking reassurance.”
- Harry Potter and the Deathly Hallows, Chapter 35 (King’s Cross)
Harry is the one who chooses the phrasing “you never killed if you can avoid it”, to contrast Dumbledore with the Dark Lord, who was certainly willing to kill those who opposed him. It’s true that Harry didn’t say “you never killed”, but wouldn’t have known all of what Dumbledore did, so he couldn’t make a statement like “you never killed” with certainty - he wouldn’t have needed to know of someone Dumbledore killed to choose that phrasing. Once Harry said it, Dumbledore wouldn’t have told Harry that he never killed at all, since he felt responsible for his sister’s death, so he wouldn’t have considered a statement that he never killed to be accurate.
Dumbledore might have accidentally killed Ariana
Dumbledore, Grindelwald and Aberforth got into a three-way duel. Ariana was hit by a stray curse, that killed her.
"It was the truth I feared. You see, I never knew which of us, in that last, horrific fight, had actually cast the curse that killed my sister. You may call me cowardly: You would be right, Harry. I dreaded beyond all things the knowledge that it had been I who brought about her death, not merely through my arrogance and stupidity, but that I actually struck the blow that snuffed out her life" -Albus Dumbledore,book 7(Deathly Hallows)
This is corroborated by his brother Aberforth, as well..
" - and I think she wanted to help, but she didn't really know what she was doing, and I don't know which of us did it, it could have been any of us - and she was dead" -Aberforth Dumbledore,book 7(Deathly Hallows)
That being said, Dumbledore hasn't killed anyone intentionally. He didn't kill Grindelwald who was the other likely candidate to have killed Ariana;he only imprisoned him. Even after Sirius' death during the battle at the Ministry, Dumbledore never killed the captured Death Eaters. He trapped them in Anti-Apparation wards, and allowed them to live.
Not after he defeated Grindelwald at least
In Limbo Dumbledore said:
I was fit only to possess the meanest of them, the least extraordinary. I was fit to own the Elder Wand, and not to boast of it, and not to kill with it. I was permitted to tame and to use it, because I took it, not for gain, but to save others from it
Meaning he presumably never killed since he's got the Elder Wand.
Probably he has not killed before that either
We do not know for sure, but the most probable situation we can imagine for him to kill would be the war with Grindelwald. And he didn't even killed Grindelwald - the most dangerous of all. More than that - during the war (that was held outside Britain) Dumbledore was evading any participation.
Years passed. There were rumors about him [Grindelwald]. They said he has procured a wand of immense power. I, meanwhile, was offered the post of Minister of Magic... But while I busied myself with the training of young wizards, Grindelwald was rising an army. They say he feared me, and perhaps he did, but less, I think, than I feared him... I delayed meeting him until finally it would have been too shameful to resist any longer. People were dying and he seemed unstoppable, and I had to do what I could. Well, you know what happened next. I won the duel.
So Dumbledore did not participate in that war the way he did in the war with Voldemort and his Death Eaters. Thus he probably didn't happen to kill anyone in that war either.