Everyday, innumerable people are found to take bath in the Holy Ganges in the hope of all their sins getting washed away. However, I have heard from my Mom many times that dips in the Ganges do not wash away sins according to Shri Ramakrishna Paramahamsa. Is this true? Do any of the biographies indicate this? If yes, what is the logic behind it?
Not exactly. Here's is what he said (Gospel of Sri Ramakrishna, Chapter 8 - available here - http://www.belurmath.org/gospel/chapter08.htm):
Efficacy of earnest japa
A Vaishnava goswami was seated in the room. The Master said to him: "Well, what do you say? What is the way?"
GOSWAMI: "Sir, the chanting of God's name is enough. The scriptures emphasize the sanctity of God's name for the Kaliyuga."
MASTER: "Yes, there is no doubt about the sanctity of God's name. But can a mere name achieve anything, without the yearning love of the devotee behind it? One should feel great restlessness of soul for the vision of God. Suppose a man repeats the name of God mechanically, while his mind is absorbed in 'woman and gold'. Can he achieve anything? Mere muttering of magic words doesn't cure one of the pain of a spider or scorpion sting. One must also apply the smoke of burning cow-dung."
GOSWAMI: "But what about Ajamila then? He was a great sinner; there was no sin he had not indulged in. But he uttered the name of Narayana on his death-bed, calling his son, who also had that name. And thus he was liberated."
MASTER: "Perhaps Ajamila had done many spiritual things in his past births. It is also said that he once practised austerity; besides, those were the last moments of his life. What is the use of giving an elephant a bath? It will cover itself with dirt and dust again and become its former self. But if someone removes the dust from its body and gives it a bath just before it enters the stable, then the elephant remains clean. .
"Suppose a man becomes pure by chanting the holy name of God, but immediately afterwards commits many sins. He has no strength of mind. He doesn't take a vow not to repeat his sins. A bath in the Ganges undoubtedly absolves one of all sins; but what does that avail? They say that the sins perch on the trees along the bank of the Ganges. No sooner does the man come back from the holy waters than the old sins jump on his shoulders from the trees. (All laugh.) The same old sins take possession of him again. He is hardly out of the water before they fall upon him.
"Therefore I say, chant the name of God, and with it pray to Him that you may have love for Him. Pray to God that your attachment to such transitory things as wealth, name, and creature comforts may become less and less every day.
The purport being that after a person washes his sins away, he should change his life to remain pure. Simply washing away your sins and continuing on in the same manner results in little gain.