This quote (with minor changes in wording) is from American writer Elizabeth Yates:
What is a good book? I suggest it is any book that enriches experience and helps us to live more understandingly, deeply, properly. Such a book has life because it gives life. It brings the reader forward to a new frontier with wider eyes, keener ears, a more responsive heart. A writer only begins a book, it is the reader who completes it; for the reader takes up where the writer left off as new thoughts stir within him. Whether that reader be child or man, the book is as good as its effect on him.
Elizabeth Yates, quoted in Ruth M. Brackbill (1955), ‘Growing with Good Books’, Christian Living: A Magazine for Home and Community, December 1955, p. 6. Also quoted in David Harris Russell (1961), Children Learn to Read, second edition, Ginn and Company, p. 422.
Brackbill gives the citation “English High Lights, March–April 1955, p. 1”, which I have not been able to find online. However, the text in bold seems to have been adapted by Yates from a sentence in her acceptance paper for the 1951 Newbery Medal, which she won for her book Amos Fortune, Free Man:
In my last year at school I wrote an essay on Browning. It was returned to me bearing a large A. Underneath the A my teacher had written, “A delightful paper to read.” That was my first real thrill in the world of letters. The path that I had been travelling so pleasantly had now begun to rise and I felt the exhilaration of air from the heights blowing across my face. The mark meant only that my facts were correct, that most of my commas were in the right places, and that no words were misspelled; but that my work had been a delight to read—that was something to feel excited about! I had a feeling that for once words had bent to my desire. I glimpsed something then that I was to see ever more clearly—that the writer only commences; it is the reader who completes.
Elizabeth Yates (1951). ‘Climbing Some Mountain in the Mind’. In Bertha Miller & Elinor Field, eds. (1955). Newbery Medal Books 1922–1955, volume I, p. 362. Boston: Horn Book.
Yates later wrote that she got the phrase from Christopher Morley:
“Why did you choose to end the story so quickly?” A question as serious as that deserves a serious and satisfying answer. When I replied I endeavored to explain how I felt about endings—that they should tell enough but not too much, that they should leave the reader stimulated but not satiated. Christopher Morley once said that “a writer only commences, it is the reader who completes the book.” Reading is communication from one mind to another, and from thence the ripples reach.
Elizabeth Yates (1967). ‘Why did you end your story that way?’. The Horn Book Magazine, December 1967, p. 709.
Yates gives no reference and I have not been able to trace it to any work by Morley, but of course it may have been a personal communication.
Tracing the origin of quotations is an activity in which no-one can be certain that they have reached the end of the trail. In this case, Yates says that she got the quotation from Morley, but for all we know Morley got it from someone else, and so on, the works containing the original versions of the quotation being lost or otherwise unavailable online. However, we are not completely in the dark:
The complete works of Samuel Johnson are online in various places, for example at yalejohnson.com and archive.org. If this quotation were really by Johnson, it would be easy to find it in these collections. (archive.org uses optical character recognition, so OCR errors make it possible for searches to miss valid instances, but this does not apply to the Yale Digital Edition.)
The sentiment of the quotation, that books are open-ended and meaning is constructed by readers, seems at variance with what we know of Samuel Johnson. If ever a writer tried to pin down meaning in precise and elegant prose, Johnson was that writer.
The evolution between the 1951 and 1955 passages above is strong evidence that Yates is responsible for the “begins a book” phrasing of the quotation.
When quotations are passed around, it is commonplace for attributions to less famous writers to be stripped and for attributions to celebrities to be substituted. Garson O’Toole describes this process:
Mark Twain, Albert Einstein, Marilyn Monroe, Winston Churchill, Dorothy Parker, and Yogi Berra are quotation superstars. Personas of this type are so vibrant and attractive that they become hosts for quotations they never uttered. A remark formulated by a lesser-known figure is attached to a famous host. The relationship is symbiotic and often enhances the popularity of both the host and the quotation. Each host attracts specific types of quotations that conform to his or her character or accomplishments.
Garson O’Toole (2017). Hemingway Didn't Say That: The Truth Behind Familiar Quotations. New York: Little A.
The stripping of attribution to Yates and the re-attribution to Johnson is in accordance with this process.
We can see the process underway in the Google Books search results. Already in 1965 the attribution to Yates has been stripped:
Or, as one author puts it, “A writer only begins a book, it is the reader who completes it […]”
Abraham Shumsky (1965). Creative teaching in the elementary school, p. 122. Appleton-Century-Crofts.
An anonymous quotation is ripe for capture by a suitable host. Here’s an attribution to Johann Wolfgang von Goethe:
The German poet Goethe once stated, “The writer only begins a book. The reader finishes it.”
Alexander Lockhart (1997). The Portable Pep Talk, p. 249. Zander Press.
The attribution to Goethe did not stick, perhaps because the quotation does not conform to his character. Instead, attributions to Johnson took over:
As Samuel Johnson so famously said, “A writer only begins a book. A reader finishes it.”
Chris Grabenstein (2015). The Island of Dr. Libris, p. 242. New York: Random House.
User ‘bukwyrm’ points out in comments that the sentiment of this quote is similar to that of a fragment by Novalis:
Der wahre Leser muß der erweiterte Autor sein. Er ist die höhere Instanz, die die Sache von der niedern Instanz schon vorgearbeitet erhält. Das Gefuhl, vermittelst dessen der Autor die Materialien seiner Schrift geschieden hat, scheidet beim Lesen wieder das Rohe und Gebildete des Buchs, und wenn der Leser das Buch nach seiner Idee bearbeiten würde, so würde ein zweiter Leser noch mehr lantern, und so wird dadurch, daß die bearbeitete Masse immer wieder in frischtätige Gefäße kommt, die Masse endlich wesentlicher Bestandteil, Glied des wirksamen Geistes.
The true reader must be an extension of the author. He is the higher court that receives the case already prepared by the lower court. The feeling by means of which the author has separated out the materials of his work, during reading separates out again the unformed and the formed aspects of the book—and if the reader were to work through the book according to his own idea, a second reader would refine it still more, with the result that, since the mass that had been worked through would constantly be poured into fresh vessels, the mass would finally become an essential component—a part of the active spirit.
Novalis (1798). Nachgelassene Fragmente zum Blütenstaub. In Novalis: Gesammelte Werke, volume 2, p. 44. Bühl-Verlag. Translated by Maraget Mahony Stoljar (1997). Novalis: Philosophical Writings, p. 45. State University of New York Press.