The following paragraphs are taken from J.T. Christian's A History of Baptists Vol. 1, Chapter 10, "The Achievements of the English Baptists," pp. 352-353. Both volumes of Christian's work are an excellent treatment of Baptist history. BTW, the bold fonts and underlines are placed there by me for emphasis.
"At a meeting held in Kettering, October 2, 1792, the Baptist Missionary Society was formed, and the first collection for its treasury amounting to £18 2s 6d, was taken up. Mr. Fuller was appointed the first Secretary, and while others nobly aided, Andrew Fuller was substantially the Society till he reached the realms of glory. Speaking of the mission to India, he says:
Our undertaking to India really appeared to me, on its commencement, to be somewhat like a few men, who were deliberating about the importance of penetrating into a deep mine, which had never before been explored. We had no one to guide us, and while we were deliberating, Carey, as it were, said, "Well, I will go down if you will hold the rope." But before he went down he, as it seemed to me, took no oath from each of us at the mouth of the pit, to this effect, that while "we lived, we should never let go the rope" (Ivimey, History of the English Baptists, IV. p. 529)."
1. William Carey was a great Baptist Missionary.
2. William Carey did not literally say anything about any rope.
3. Andrew Fuller was a great Baptist Theologian/Baptist Missionary Society Secretary.
4. Andrew Fuller was using an illustration to make a point.
5. If any one's name should be attributed to the "rope quote" it should be Andrew Fuller.