In "The Life and Lies of Albus Dumbledore" Rita Skeeter mentions that Albus Dumbledore promoted Muggle rights:

Harry Potter and the Deathly Hallows, chapter 18:

Astonished and appalled though his many admirers will be, this letter
constitutes the Statute of Secrecy and establishing Wizard rule over
Muggles. What a blow for those who have always portrayed Dumbledore as
the Muggle-borns' greatest champion! How hollow those speeches
promoting Muggle rights seem in the light of this damning new
evidence! How despicable does Albus Dumbledore appear, busy plotting
his rise to power when he should have been mourning his mother and
caring for his sister!

What are these Muggle rights that Dumbledore promoted?

Dumbledore believed Muggles deserved to be considered.

When Dumbledore mentions the disappearance of Frank Bryce, a Muggle, he also says that he regrets that the Ministry doesn’t consider Frank’s disappearance important because he’s a Muggle.

“And there was a third disappearance, one which the Ministry, I regret to say, does not consider of any importance, for it concerns a Muggle. His name was Frank Bryce, he lived in the village where Voldemort’s father grew up, and he has not been seen since last August. You see, I read the Muggle newspapers, unlike most of my Ministry friends.”
_{^{- Harry Potter and the Goblet of Fire, Chapter 30 (The Pensieve)}}

This indicates that Dumbledore himself does believe the disappearance of Muggles to be important to consider, and that he reads the Muggle newspapers implies he’s interested in Muggle events.

He refused to ban stories with pro-Muggle messages in Hogwarts.

Though Dumbledore has been asked to remove certain stories such as “The Fountain of Fair Fortune” from the Hogwarts library because of their messages on Muggles, he’s refused.

“My refusal to remove the book from the library was backed by a majority of the Board of Governors. I wrote back to Mr. Malfoy, explaining my decision:

So-called pure-blood families maintain their alleged purity by disowning, banishing, or lying about Muggles or Muggle-borns on their family trees. They then attempt to foist their hypocrisy upon the rest of us by asking us to ban works dealing with the truths they deny. There is not a witch or wizard in existence whose blood has not mingled with that of Muggles, and I should therefore consider it both illogical and immoral to remove works dealing with the subject from our students’ store of knowledge.⁸”
_{^{- The Tales of Beedle the Bard}}

From his letter, Dumbledore clearly doesn’t believe that being pure-blood means anything, and he doesn’t consider the idea of wizards marrying Muggles something that should be banned.

He likely made pro-Muggle decisions in his positions of authority.

Dumbledore held many positions of authority in the wizarding world, such as being the Supreme Mugwump of the International Confederation of Wizards and the Chief Warlock of the Wizengamot.

“Headmaster: Albus Dumbledore

(Order of Merlin, First Class, Grand Sorc., Chf. Warlock, Supreme Mugwump, International Confed. of Wizards)”
_{^{- Harry Potter and the Philosopher's Stone, Chapter 4 (The Keeper of the Keys)}}

In those positions, he could make and influence decisions about the laws of the wizarding world.

“Dumbledore’s innumerable contributions to the store of wizarding knowledge, including his discovery of the twelve uses of dragon’s blood, will benefit generations to come, as will the wisdom he displayed in the many judgements he made while Chief Warlock of the Wizengamot.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 2 (In Memoriam)}}

It’s likely he made pro-Muggle decisions in these positions, especially considering that the wizarding world knows him as a champion of Muggle rights, whatever their own personal opinions of Muggles - those who don’t know him personally would need some reason to believe this.

Those against Muggles believe Dumbledore champions them.

Also, those who oppose Muggles consider Dumbledore someone who fights for them, and they’d know who their opponents are. The Dark Lord, who intends to eventually raise wizards up to rule over the Muggles, calls Dumbledore the champion of commoners, of Mudbloods and Muggles.

“And I answer myself, perhaps they believed a still-greater power could exist, one that could vanquish even Lord Voldemort … perhaps they now pay allegiance to another … perhaps that champion of commoners, of Mudbloods and Muggles, Albus Dumbledore?”
_{^{- Harry Potter and the Goblet of Fire, Chapter 33 (The Death Eaters)}}

The Dark Lord almost certainly has some basis for considering Dumbledore a strong opponent of his cause in this way - the Dark Lord would have to know who he’d be fighting against.

He was described as a determined supporter of Muggle rights.

In the eulogy Dumbledore’s friend Elphias Doge wrote about him, Doge described Dumbledore as someone who never seemed against Muggles and determinedly supported Muggle rights.

“They could not have been more mistaken: as anybody who knew Albus would attest, he never revealed the remotest anti-Muggle tendency. Indeed, his determined support for Muggle rights gained him many enemies in subsequent years.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 2 (In Memoriam)}}

Though Elphias Doge was Dumbledore’s friend so was more likely to look kindly on him and not say anything that could be considered bad about his dead friend, it’s unlikely he’d have entirely made up that Dumbledore had supported Muggle rights, especially considering the other evidence.