I think you are almost certainly quoting that out of context. Franklin’s contribution was significant, and unfairly minimized in some of the early accounts (partly because of the casual sexism prevalent in the era, to which Watson may have been particularly prone, but probably even more because of the bad relationship, based largely on misunderstandings, she had with her nominal boss, Maurice Wilkins), but the importance of her role in the eventual understanding of DNA has subsequently been hugely exaggerated (for rather obvious ideological reasons) in some of the more recent accounts. Both Watson and (the slightly less assholish) Crick may have felt sufficiently shamed by the lack of due credit she received early on to rather exaggerate her importance in some of their later remarks, such as the one you quote. However, to say or imply that hers was the main basis of the discovery, is at least as misleading as the original implications that she had rather little to do with it, or was more obstructive than helpful.
Personally I think she deserved a share of the Nobel rather more than Maurice Wilkins did, and she would probably have got it (perhaps instead of Wilkins) had she still been alive by the time the importance and correctness of the double-helix DNA model was sufficiently established for a Nobel to be appropriate. However, people who imply that her contribution to the discovery was equal to, or (as some seem to want to make out) greater than that of either Watson of Crick, or that she was in any sense “cheated” of the Nobel prize, are talking out of their ass.
In fact, the person who almost certainly deserved the most credit, and most deserved a Nobel, for the DNA discovery, after Watson and Crick, was Erwin Chargaff. His contribution to their discovery was, in my judgement, considerably more crucial than Franklin’s, but he was an asshole too, in his own way, and he became extremely bitter that Watson and Crick received so much more fame and kudos than he did. If anyone was “cheated” of their due share of the Nobel for DNA, though, it was him rather than Franklin. (But as a man, and something of jerk, he does not have an army of revisionist feminist historians out there pleading his case.)
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As for Watson, the science that he made his name with is still good, but, through his racism and so forth he has forfeited the respect often accorded to Nobel prize winners, and “great scientists” in general, as intellectuals with possibly wise things to say about areas of science outside their specialty, or even about politics, philosophy, or human society in general. Watson is not disgraced as a scientist, just old and washed up, but he is disgraced as a public intellectual and pundit, which is a role that top scientists often aspire to, and sometimes even fill quite well, when they are past their prime as actual scientists.