Some of the actors mentioned upthread, Jerry Mathers, Jay North, and Johnny Crawford, were children when they started out in TV and are only in their 60s today. I’d say it’s very reasonable to expect them to still be alive. I’d say it’s only remarkable when someone who has passed 80 or 90 is still with us.
Nanette Fabray. She was on the successor show to “Your Show of Shows”, called “Caesar’s Hour”.
June Foray is still around, and is still doing it. She’s been voicing animated characters since before TV, but I think she counts.
Adam West was showing up for episodes of The Philco Television Playhouse in '54, long before earning paychecks on all those TV Westerns in the late '50s: heck, he portrayed Doc Holliday on Sugarfoot, and Doc Holliday on Colt .45, and Doc Holliday on Lawman, and otherwise popped in for an episode of Cheyenne or Bronco or Maverick at need.
Efrem Zimbalist and Edd Byrnes (77 Sunset Strip) are still around.
Some '50s child stars, like Frank Bank, Stanley Fafara, and Annette Funicello, are not.
Barbara Hale was playing Della Street on Perry Mason well before '58, and before that was on every series from Playhouse 90 to Science Fiction Theater.
Oh, and Martin Landau’s still working, more than sixty years after his first TV credit. And how about Eli Wallach, who made his television debut way back in the '40s before all his work on Playhouse 90 in the '50s?
That’s one of the half-a-dozen TV shows where Michael Caine got work pre-'58!
Still, it’s kind of remarkable that Christopher Walken was a child actor on television back in '53, because who the heck knew that?
She was mentioned in the O.P.
Clint Eastwood.
John Saxon isn’t set to turn 80 until next year, but first appeared on TV when he was in his twenties, back in '55, which is still pretty good. (He was still in his twenties when he appearing on everything from Gunsmoke to General Electric Theater, at that.)
As per IMDB, Jack Nicholson’s first screen credit is '56, in the Matinee Theater TV series, not long before landing roles on Tales of Wells Fargo and Sea Hunt.
Before he was Oscar Goldman on The Six Million Dollar Man, Richard Anderson had a recurring role as Ricardo del Amo on Zorro back in 1958; before that, he spent 1957 appearing on Zane Grey Theater and Schlitz Playhouse; before that, Captain Midnight and Chevron Hall Of Stars.
William Schallert, a prolific character actor is 91. He started working in TV and movies in 1949 and is probably best known as Martin Lane in “The Patty Duke Show.” “Star Trek” fans will remember him as under-secretary Nilz Baris in “The Trouble With Tribbles.”
I saw Jack Nicholson on an Andy Griffith Show rerun from around 1962. He was arrested for speeding in Mayberry IIRC. He started his screen career as a lawbreaker early.
William Roache misses your cut by 2 years, but he has been playing Ken Barlow continuously since 1960.
Don Hastings started in TV in 1947, and appeared in Captain Video in 1949. He went into soap operas in the 1950s and was a regular on As the World Turns until it went off the air in 2010.
Back in '52, William Daniels was playing John Quincy Adams on television, because of course he was. He then appeared in Robert Montgomery Presents and Justice and Armstrong Circle Theater by '57, followed by decades of better-remembered stuff.
Robert Loggia also qualifies: before getting his big break as Elfego Baca in '58, he picked up a recurring role in '57 as Joe Crane on Studio One In Hollywood.
Joanne Woodward acted on literally dozens of TV shows before 1958, when she famously appeared with Paul Newman on Playhouse 90 (along with Daryl Hickman, who had plenty of pre-1958 work under his belt: The Lone Ranger, Perry Mason, Pulitzer Prize Playhouse, Alfred Hitchcock Presents, General Electric Theater, Men Of Annapolis, and et cetera).
Noel Neill and Jack Larson each racked up pre-1958 TV work even aside from their signature roles as Lois Lane and Jimmy Olsen in The Adventures of Superman.