"The King of Swing," Benny Goodman, was buried yesterday in a plan wooden casket next to his wife, who died in 1978. The ceremony, in Stamford, Conn., was private and attended only by 40 close family and friends. It was a nonsectarian cermony, at his wish; his two daughters, Benjie and Rachel, and his three stepdaughters read from the Bible at graveside, according to a neighbor who attended the cermony.

On Saturday reaction to his death Friday at age 77 continued. President Reagan said he and First Lady Nancy Reagan were saddened.

Said Reagan: "Benny's swing music came to symbolize the new energy of 20th-century America and it helped us all through the Great Depression. It was a new and uniquely American sound that was as fresh as the land that inspired it. We miss Benny but will forever remember his contribution to American life and music."

Longtime colleague Frank Sinatra said "Working with Benny Goodman wasn't a job; it was an experience." Lionel Hampton, one of the first blacks to play in a formerly all-white band when Goodman integrated his band in the 1930s, said, "He was really a great man, he was a godsend to the world." 'Carousel' Notes

The irrepressible comic actor Jack Gilford, who was to play the Starkeeper in the new production of Rodgers and Hammerstein's "Carousel," opening at the Kennedy Center this week, had to leave the show because he has scheduled some emergency but "minor" surgery. Veteran Broadway actor Milo O'Shea came into town Saturday to replace him. The production opens for preview performances tonight. Thursday is opening night.

Going along with the musical's theme, Kennedy Center Chairman Roger Stevens is hosting a different kind of opening-night party for the company. Late that evening, the cast and invited first-nighters will attend an outdoor clambake at the Carousel on the Mall near the Smithsonian Castle. There aren't many people on the Mall at that time of night. End Notes

Now that the comic strip "Brenda Starr, Reporter" is about to become a major motion picture, director Robert Ellis Miller ("The Heart Is a Lonely Hunter") has found his Starr. He will introduce longhaired, starry-eyed look-alike Brooke Shields, who will play the Flash's star reporter, at a press conference in New York tomorrow morning. Miller has promised Shields that her work will be completed in time for her to get back to classes at Princeton in the fall . . .

CIA Director William Casey has often been heard to grumble about the statue of Revolutionary War hero Nathan Hale, which stands at the entrance of the CIA in Langley. He describes Hale as having "fouled up the only mission he was ever given." The British caught him spying and hanged him. Casey is about to change things. He has ordered a new statue of his own hero. According to Newsweek magazine, there will be a statue of Gen. William (Wild Bill) Donovan someplace at the CIA. Donovan was the founder of the precursor of the CIA, the Office of Strategic Services, Casey's unit in World War II . . .

British Prime Minister Margaret Thatcher's son Mark is still having problems keeping an apartment in Dallas, where he lives. Since his mother supported the U.S. bombing of Libya in April, he was forced to leave one apartment building because residents feared a terrorist attack. He is under 24-hour State Department protection, and now another apartment building where he is staying wants him out. Former senator John Tower grumbled that a memo being circulated in the building seeking Mark Thatcher's ouster "makes us Texans look like a bunch of yellow-bellies" . . .

Lee Iacocca is a man who gets even. It must be especially satisfying to him to have purchased a seven-bedroom Georgian-style home in Grosse Pointe Farms, Mich., that once belonged to Henry Ford II. He's the man who fired Iacocca in 1978, reportedly saying, "I just don't like you." The price of the house was not revealed but is estimated at between $800,000 and $1.5 million. It's probably worth every penny to Iacocca . . .

Ursula Meese picked the right horse at Laurel Race Track yesterday. She was there chairing a fundraiser for the National Capital Chapter of the Multiple Sclerosis Society, and she put her money on Here's a Toast in the first race. She didn't take too much of a gamble, since the horse was a 2-1 favorite. But when it came in she announced that since it was Father's Day, she would donate the winnings to her husband, Attorney General Edwin Meese, who was there with her . . .