I remember hearing about some study finding that women spoke an average of 20,000 words per day while men spoke only 7,000. It's hard to get an accurate count of something like that, so I figured it was based on a small non-representative sample. According to a recent story in the New York Times, it probably wasn't based on any kind of sample: it seems that the numbers were just invented.
The story went on to suggest that "this stereotype may dovetail with the idea that what women have to say isn’t important — that it’s 'fluff,' and that "such sterotypes [may] make women less likely to speak up, or men less likely to hear them..." I had a different impression--that it was associated with the idea that women had more "emotional intelligence" than men. A 2000 Gallup survey contains has some relevant information. It listed a number of characteristics and asked if each was "generally more true of men or more true of women" (people could volunteer that there was no difference). It also asked if "the country would be governed better or governed worse if more women were in political office" and "if you were taking a new job and had your choice of a boss would you prefer to work for a man or a woman?" The characteristics were: aggressive, emotional, talkative, intelligent, courageous, patient, creative, ambitious, easy-going, and affectionate. 77% say women are more talkative, 11% say men, and 10% say no difference, which is about the same as when the question was first asked in the 1940s.
Opinions about which sex is more intelligent, courageous, and patient help to predict opinions about whether more women in office would mean better or worse government. Opinions about which sex is more intelligent, courageous, and easy-going help to predict preferences about a man or woman as boss. That is, people who see women as more intelligent, courageous, patient, or easy-going are more likely to think that the country would be governed better or prefer a woman as a boss. The others, including talkative, do not have a statistically significant relationship. (For what it's worth, the estimates for talkative are positive --favorable-- with t-ratios of 1.3 and 1.0).
You might wonder if belief that woman are more talkative is part of a pattern, going with negative views about women's intelligence, courage, etc. It has a significant negative association with courageous--that is, people who see women as more talkative tend to see men as more courageous--but not with views about which sex is more intelligent, easy-going, or patient. Overall, the correlations with opinions about other qualities were low.
So in conclusion, the stereotype doesn't seem to matter much either way.