“This is a family newspaper” is a frequently printed newspaper admonition to keep it clean—no obscenities are allowed in print. The phrase is often given as “We would say what we really feel about him/her/it, but this is a family newspaper.”  “March came in like a lamb and has gone out like—however, this is a family newspaper” was cited in print in 1881.
 
Television and radio programs have a similar saying in “This is a family show.”
 
 
1 April 1881, Daily Illinois State Journal (Springfield, IL), “Local Notes and Personals,” pg. 6, col. 3:
March came in like a lamb and has gone out like—however, this is a family newspaper.
 
15 August 1888, Boston (MA) Daily Globe, “Brief Brieflets,” pg. 4, col. 6:
We have received a communication from Hon. John J. Ingalls, but we want it to be distinctly understood that this is a family newspaper.
 
24 January 1898, Bay City (MI) Times-Press, “Sisterly Cities,” pg. 5, col. 2:
Let us discuss New York society. But, come to think of it, we cannot. This is a family newspaper.—Chicago Times-Herald.
 
12 February 1913, Flint (MI) Daily Journal, ‘State press Notes,” pg. 4, col. 3:
Our idea of zero temperature shall remain unexpressed. This is a family newspaper.—Pontiac Press Gazette.
 
18 April 1965, Boston (MA) Globe, “Across the City Desk” by Joseph Dinneen Jr., pg. 47:
The best answer I can give you Mr. Bailey is this: This is a family newspaper, you know.
 
Google News Archive
26 April 1965, Ottawa (Ontario) Citizen, “Canadiens lose and Blake mad at the world” by Jack Kinsella, pg. 15, col. 3:
Well, I can appreciate that all lovers of drama and the stage would dearly love to hear some of the other things which were said in the ensuing monologue, but really, most of them would be irrelevant, and a lot of them unprintable in view of the fact this is a family newspaper.
   
The Internet Movie Database
Memorable quotes for
All the President’s Men (1976)
John Mitchell: [on phone] You tell your publisher, tell Katie Graham she’s gonna get her tit caught in a big wringer if that’s published.
Ben Bradlee: [later] He really said that about Mrs. Graham?
Carl Bernstein: [nods]
Ben Bradlee: Well, I’d cut the words “her tit” and print it.
Carl Bernstein: Why?
Ben Bradlee: This is a family newspaper.
 
23 May 1976, New York (NY) Times, “Music View: Memories of an Acoustical Dud” by Harold C. Schonberg, pg. X23:
But this is a family newspaper, and it never appeared.
 
Google News Archive
7 November 1977, Pittsburgh (PA) Post-Gazette, Ann Landers syndicated column, pg. 13, col. 2:
Sorry I had to change the first letter in the last word but this is a family newspaper.
     
Los Angeles (CA) Times
Goldberg: Rage against the TSA machine
The agency’s airport screening policy seems to be that you hassle everyone equally, even a 95-year-old with leukemia.
June 28, 2011|Jonah Goldberg
(...)
I’d tell you how thorough the TSA search was of blogger and advice columnist Amy Alkon (who collects such tales), but this is a family newspaper. Suffice it to say, your government left nothing to chance.
   
Albany (NY) Times Union
Diamond debate
Published 12:14 a.m., Tuesday, July 12, 2011
(...)
JORDAN: What!? Why? His back still hurt from trying to bury Kevin Gregg with his left hand of doom? I wish I could repeat what Fake Derek Jeter tweeted about that fight. But this is a family newspaper. Go Google it, enterprising youths.