Anecdotal and anomalous statistical evidence aside, it's tough to find proof that more babies are born on a full moon. The topic has been studied pretty extensively, though, and it's very easy to find evidence disproving the connection.
Here are just a handful of the scientific studies that have found no connection between the full moon and birth rates:
1957: Looked at a series of days with abnormally high numbers of births and tried to correlate them with full moons. No correlation was found [source: Shulman].
1987: Looked at U.S. birth rates by decade and found no correlation between full moons, birth rates or conception rates [source: Shulman].
1996: Looked at 100 previous studies on lunar effects and found no statistically significant proof of the moon's effect on birth, violence, suicides, major disasters or a dozen other supposedly lunar-connected phenomena [source: Skeptic's Dictionary].
1998: Looked at 3,706 births and found that "scientific analysis of data does not support the belief that the number of births increases as the full moon approaches, therefore it is a myth not reality" [source: ScienceDirect].
2005: Looked at 564,039 births in North Carolina between 1997 and 2001 and found "no predictable influence of the lunar cycle on deliveries or complications" [source: PubMed].
2006: Looked at births over a 28-year period in Australia and found that "full moons are not associated with any significant change in the number of conceptions, births, or deaths" [source: Gans].
If there's so much evidence against the lunar effect, why is the belief so widespread? For pretty much the same reasons why lots of other old wives' tales or urban legends pick up a strong following: It's easy to believe.