Dennis Mangan links to a study finding positive relationships between greater reproductive success and higher extraversion, lower conscientiousness, and lower openness to experience, as well as a more attenuated positive correlation with higher agreeableness.
The GSS dipped its toe in big five waters back in 2006, asking survey participants ten questions, two per trait (which is suboptimal, but we work with what we have). The mean number of children among respondents aged 30 or older by each trait (bifurcated for simplicity):
Extraverted -- 2.15
Introverted -- 2.19
Agreeable -- 2.38
Disagreeable -- 2.22
Conscientious -- 2.27
Unconscientious -- 1.98*
Stable -- 2.28
Neurotic -- 2.34
Open to experience -- 2.19
Closed to experience -- 2.17
Some marginal suggestive support for agreeableness but not for extraversion, conscientiousness, or openness to experience.
GSS variables used: BIG51A(1-2)(4-5), BIG51B(1-2)(4-5), BIG51C(1-2)(4-5), BIG51D(1-2)(4-5), BIG51E(1-2)(4-5)
* Only 8 valid cases--the others were in the 3- or 4-digits range--unfortunately, the conscientiousness questions are worded in such a way that a respondent really has to make himself out to be a dirt bag to categorize as not being conscientious; that is, he has to describe himself as someone who is lazy and incapable of doing a thorough job on things.