Seems to have high merit as a motivation to think and speak with civility.
Spencer W. Kimball coined the phrase:
"Profanity is the effort of a feeble brain to express itself forcibly."
in October of 1974, while serving as President of the Church of Jesus Christ of Latter-day Saints.
This same speech was later published in November of the same year, in the General Conference Report issue of the Ensign magazine.
A Google Books search finds a number of earlier expressions of the sentiment (attributed to Spencer Kimball in 1974) that profanity is evidence of a weak mind.
Instances from the period 1866–1919
From "Profanity.—A Christian Gentleman," in American Phrenological Journal and Life Illustrated (January 1866):
Profanity is simply an evidence of a vulgar mind, sadly perverted, whose influence must inevitably be bad. We pity and loathe the weak mind which falls into a habit so silly, so impotent, and so foolish.
From Henry Gibson, Catechism Made Easy: A Familiar Explanation of the Catechism of Christian Doctrine, second edition, volume 1 (1882):
Some children are so foolish as to imagine that it makes them look big and like men if they season their speech with profane words. This is the mark of a weak mind ; for a modest, simple, and candid way of speaking is one of the most pleasing ornaments of youth.
From a letter to the editor by a Philadelphia dentist printed in Dental Items of Interest: A Monthly Journal of Dental Art, Science and Literature (1897):
The "trade journal" epithet is somewhat like a profane utterance which has been said to "be the strong language of a weak mind."
From "Profanity," in Massachusetts Reformatory, Our Paper (November 24, 1906):
Profanity never indicates high station or a well bred person. It is an evidence of low breeding, of limited ideas and a weak mind. The societies of men and women who are working to combat the influence of profanity will do a good and noble work if they can raise the American people from the title that names them the most profane people in the world.
From "The Physiology of Profanity," in Medical Consensus (March 1908):
Of course extremes are always harmful and excessive exhibition of anger is pathological, just as excessive hilarity is pathological. Moreover the foregoing contention [that the use of expletives under appropriate circumstances is physiologically and emotionally healthful] does not apply to wanton profanity, which has no physiological basis other than a weak mind or coarse associations.
But watch the man who never smiles, and watch the man who NEVER swears. They are both unbalanced—abnormal.
From "Plea for Cleaner Speech," in The Barbed Wireless (September 1918), reprinted in The Red Cross Magazine (June 1919):
There is nothing manly nor humorous in profanity or obscene language. It is an indication of a weak mind rather than a strong one. Cut it out!
Instances from the period 1947–1963
From Paul Holdscraft, Cyclopedia of Bible Illustrations (1947) [snippet]:
1078. Profanity defined.—Someone has given this definition of profanity: "The effort of a feeble mind to express itself forcibly."
From a series of aphorisms in The Defender Magazine: 1947–1948 (1948[?]) [snippet]:
Truth bottled up and slyly choked, becomes a lie though never spoken.
Profanity is the effort of a weak mind, trying to express itself forcibly.
No law has ever been made, that will keep a man from making a fool of himself.
From a series of aphorisms in Franciscan Message (1951) [snippet]:
Being born with a silver spoon in his mouth has caused many a boy to have difficulty in stirring for himself later on in life.
If we could learn to enjoy the things we dislike, perhaps we would have another way of enjoying life.
Profanity: a strong way of expressing a weak mind.
From Canadian Saturday Night: A Magazine of Business & National Affairs, volume 71 (1956) [combined snippets]:
Interviews are often broken by telephone calls and knocks on the door. The [highways] minister looks and sounds vaguely like a movie gangster as he barks into the phone. His voice is loud but melodious. He sprinkles his colorful public speeches with quips and adages—"many have the right aim in life but never seem to get around to pulling the trigger"; "profanity is evidence of a weak mind trying to express itself forcibly".
From a series of aphorisms in Oysters and Politics (1963) [snippet]:
The perfect example of arrested motion is a woman entering her thirties.
Profanity is a strong way to express a weak mind.
It just seems to happen that by the time most of us get a little money to burn "the fire goes it."
The sentiment that profanity is the effort of a weak mind to express itself forcibly (or forcefully) goes back at least 150 years, and instances of similar wordings of the idea as an aphorism go back to 1947 at the very latest, and arguably to 1897. As these instances show, giving Spencer Kimball credit for having coined the phrase on the strength of his having used it in October 1974 is not justified.
Spencer W. Kimball apparently said it in the form here: Spencer W. Kimball > Quotes
“Profanity is the effort of a feeble brain to express itself forcibly.”
I assume that this has been taken from his writings.
It seems to be all over the internet in a variety of forms, adapted by each speaker to fit their own needs.
The earliest attribution to Kimball (that I can find) is 22nd of October 2002, here.