Since you have added the books tag, I will avoid the show heresy and answer this from the books.
###What do Others aka White Walkers wear?
In books, others do not run around naked. They are dressed in a special magical armor which camouflages them into their surroundings.
From wiki:
"Had many talks with George. He told me of the ice swords, and the reflective, camouflaging armor that picks up the images of the things around it like a clear, still pond. He spoke a lot about what they were not, but what they were was harder to put into words."- Tommy Patterson (Comic book artist for the series)
Also:
A shadow emerged from the dark of the wood. It stood in front of Royce. Tall, it was, and gaunt and hard as old bones, with flesh pale as milk. Its armor seemed to change color as it moved; here it was white as new-fallen snow, there black as shadow, everywhere dappled with the deep grey-green of the trees. The patterns ran like moonlight on water with every step it took.
AGOT - Prologue
So while we may not know what exactly do they wear due to reflective nature of their garb, we know they wear an armor so presumably some other garment underneath as well. Therefore, The idea that they run around naked or with minimal clothing is absurd.
As far as books are concerned, they always wear armor (Except maybe when it is a nice sunny day, then I presume they go sunbathing on the Frozen shore).
###Why did the Show depict them as a naturalist club in the beginning?
