According to Eugene Maleska, former crossword puzzle editor for The New York Times, Arthur Wynne first introduced simple crosswords to the Fun page of The New York World Sunday Magazine in 1913. NPR did a 100 year anniversary special about it and reproduced Wynne's first newspaper crossword

They became a fad in the 1920s and then really took off in the 1930s when puzzle writers started using more punny clues rather than dictionary definitions.

Maleska first submitted puzzles to The New York Herald Tribune in 1940. At that time the New York Times still wasn't running a daily crossword puzzle.

Crossword puzzles and newspapers or magazines seem like a natural fit. In any literate audience there will be a number of people who love word play and the information-dense format is a great fit for print. It's really surprising that comics like The Yellow Kid predate them by almost twenty years.

Here are a bunch of scans of the Fun pages.

To add to /u/mkwiat's summary:

The crossword fad of the 20s was really huge. Today, Simon & Schuster is a mammoth publishing company, one of the biggest worldwide. It got its start publishing crossword puzzle books.

The New York Public Library complained in 1924 that ""The latest craze to strike libraries is the crossword puzzle," and that when "the puzzle 'fans' swarm to the dictionaries and encyclopedias so as to drive away readers and students who need these books in their daily work, can there be any doubt of the Library's duty to protect its legitimate readers?" Source

So the crossword became a hit shortly after Arthur Wynne's first puzzle of 1913. I'm most familiar with the Times puzzle, so I'm going to address that: the New York Times was late to the game (probably resisting what seemed at first to be mere frivolity); the puzzle first started in 1942.

It was, of course, the middle of WWII, and a memo at the paper less than 2 weeks after Pearl Harbor specifically referred to the need for diversion in those (literally) dark evenings (due to the blackouts):

"We ought to proceed with the puzzle, especially in view of the fact that it is possible that there will now be bleak blackout hours -- or if not that, then certainly a need for relaxation of some kind or other. (That, in turn, raises the question of whether we ought not to have a crossword puzzle in the daily also [NB: the Times did indeed end up only running a Sunday puzzle at first] -- but that's a different story.) Source

The Times of course, intended to elevate this diversion a bit for its audience; the memo goes on to state ""We ought not to try to do anything essentially different from what is now being done -- except to do it better."

The first crossword editor, Margaret Farrar, also stated (same article) that ""The great majority of puzzle solvers want a large, challenging, rather hard puzzle, with terse dictionary definitions plus occasional literary, historical and news references." So from the start the Times puzzle was establishing itself as more literary, more difficult, than the general run of puzzles out there. This has continued to the present day; the current editor, Will Shortz, has said ""What I especially like about being at The Times is the audience. The Times undoubtedly has the most intelligent, best educated group of solvers in the country, so I can presume a level of culture and solving skill that I couldn't anyplace else."

The Times puzzle was helped along by the fact that the paper's publisher, Arthur Sulzberger, was enough of a fan, to write a puzzle for the paper himself.

The puzzle has evolved as our culture has, but slowly: when Shortz took over in 1993, he said his changes to the puzzle were modest:

Succeeding the late Eugene T. Maleska, Mr. Shortz made modest modifications to the crossword, "but nothing radical." Constructor bylines were added to the daily puzzles; previously the contributors had labored anonymously. Also, the cultural references were broadened, with such modern subjects as movies, television and rock and roll being added to the classic puzzle vocabulary. "And I do like riddles, so there are more tricks and ambiguities in the clues now," he said. Source

EDIT TO ADD: Sudoku really became a craze starting in London about 2002-2003 as I recall. US publishers noticed the London bestseller list was full of "su-do-ku" books (spelling wasn't wholly consistent then) and brought them over here, where there was briefly a craze to rival the crossword craze of the 20s. (source: I published Sudoku books at a major publisher at the time). So those specific puzzles are quite recent; Ken Ken, in terms of popular appearance, postdated Sudoku. The idea of other puzzles besides the "main" crossword on the page, however, goes way back; Margaret Farrar at the Times envisioned what would become the "Second Sunday puzzle" (i.e., the Acrostic or other non-crossword word puzzle) from the start: "The smaller puzzle, which would occupy the lower part of the page, could provide variety each Sunday," she continued. "It could be topical, humorous, have rhymed definitions or story definitions or quiz definitions. The combination of these two would offer meat and dessert, and catch the fancy of all types of puzzlers."