As one who formerly (in ignorance) used to promote the same argument (not being a Jehovah's Witness, but somewhat associated with the Arian theology), the argument is not that the Greek text was corrupted, but rather, that the comma in the English translation should be placed after "today" rather than before it. They are able to make this argument because the original Greek manuscripts, and those for centuries thereafter, lacked any sort of punctuation as well as miniscular letters. For example, the Codex Sinaiticus is dated to the 4th century A.D. The following is the Greek text of Luke 23:43 therein:
As you can see, there are no miniscular letters and, more importantly, no punctuation.
The notion that the Lord Jesus Christ did not go to Paradise (whether it be a place in heaven or in a compartment of Hades) that same day via his soul is important to Jehovah's Witnesses who believe in "soul sleep" or "mortalism." Succinctly stated, they assert that man does not have an [immortal] soul that continues to exist after the death of the body. Rather, man is a soul,(1) and as the dead are said to sleep upon their death, then the soul (man himself) also sleeps.
The Watchtower Online Library entry on "sleep" states,
Death Compared With Sleep. There is evidence that people sleep in cycles. Each cycle is made up of a deep sleep followed by a lighter sleep. During periods of deep sleep it is very difficult to awaken a person. He is completely unaware of his surroundings and the things that may be occurring about him. There is no conscious activity. Similarly, the dead are “conscious of nothing at all.” (Ec 9:5, 10; Ps 146:4) Therefore death, whether that of a man or of an animal, is like sleep. (Ps 13:3; Joh 11:11-14; Ac 7:60; 1Co 7:39; 15:51; 1Th 4:13) The psalmist wrote: “From your rebuke, O God of Jacob, both the charioteer and the horse have fallen fast asleep.” (Ps 76:6; compare Isa 43:17.) Were it not for God’s purpose to awaken persons from the sleep of death, they would never wake up.
Therefore, since the soul of Jesus, i.e. himself, was dead and buried three days and nights in the grave, the Lord Jesus Christ could not have went anywhere during the interim, including Paradise. If this be the case, they reason, the comma could not logically be placed before the word "today" in the English translation. Furthermore, they assert that those modern Greek texts which place the comma between σοι and σήμερον are simply the result of scribal bias.
To summarize, Jehovah's Witnesses base their assertion on the fact that the original Greek manuscripts lacked punctuation and the placement of the comma by some scribes or translators after the Greek word σήμερον or English word "today" is simply a result of the scribe or translator's bias. Because man is a soul, and man sleeps upon his death, then the soul of our Lord Jesus Christ could not have went anywhere while he was dead for those three days and nights in the grave. This is the summary of their argument.
As you did not request a refutation thereof, I have not included one. But, in the event you edit your original question and request one, I would have no problem including one.
(1) They emphasize Gen. 2:7 for support of this assertion, which states that man became a [living] soul and not that man has a soul.