His crew was very close to an actual mutiny on October 10th 1492, as can be gathered from Columbus' own writings, and from Gonzalo Fernández de Oviedo, who became a good friend of Columbus' sons and of Vicente Yáñez Pinzón (one of the captains of the expedition). If we only consider the annotation from Columbus' log on October 10th, the situation did not look so dire. I quote:

Here the people could not bear it anymore, they complained about the long voyage. But the Admiral humoured them as best as he could, giving them good hope of the profits they could have. And he added that complaining was out of place, for he had come to the Indies, and so would he continue until finding them with the help of Our Lord.

This only indicates that morale was very low, but not that people were close to a mutiny. We need to go to February the 14th 1493 to get a solid grip on how dire the situation was:

For God had saved him in the journey [to the Indies], when he had more reasons to fear the sufferings he had with the sailors and people he carried, all of who with one voice were determined on turning against him and rebelling, raising their protests, and the Eternal God gave him courage against them, and many more things of great wonder that God had shown on that first voyage

Those protests are much more detailed in Gonzalo Fernández de Oviedo's General and Natural History of the Indies, book II, chapter V. Oviedo, having met Vicente Pinzón, and many more people that had gone to the Indies on the first journey, gather as many testimonies as he could, and those were relatively close in time as to have vivid memories. Let us not forget that Vicente Pinzón died in 1514, so Oviedo must have met him somewhere between 1505-1508, some 12-15 years apart from the voyage, which was something memorable. I quote Oviedo:

Those that were with him started to lose heart, and wanted to go back, and fearing the way they were going they started to murmur on Columbus' wisdom, and of his foolhardiness. And the people and captains started to mutiny, for each hour fear grew larger in their hearts, and the hope of finding the land they searched diminished. So, they publicly and shamelessly told him that he had cheated them, that he had led them to their doom, and that the King and Queen had been wrong and cruel in trusting a man like him, and trusting a foreigner who did not know what he said. So far had arrived the situation, that they assured him that if he did not turn back the ships, they would so do against his will, or toss him into the sea.

The intervention of the Pinzón brothers, who were the natural leaders of the region the sailors were from, was vital in not having Columbus thrown overboard. Columbus did have some intervention, saying that he was there with them, showing his compromise with the enterprise. However, as many complaints came from the fact that Columbus was a foreigner whose experience was unknown to the seafarers of Palos and Moguer, the fact that Martín Pinzón (a veteran captain of good name), and Vicente Pinzón (still young but with an impressive record at sea including several acts of piracy at ages 15 through 18) vouched for him, helped in defusing the situation.