My guess would be Clinton, Bush (W), or Obama. Although the birth rate is down, the growth of population has caused an increase in the number of babies being born. During the Eisenhower administration the population in 1953, when he was first elected, was 155 million. When he left office in 1961 the population was 179 million. Today the population of the US is roughly 320 million, double what it was in 1953. So, even though the birth rate is down, the number of births, due to double the number of people, has been way up starting with the Clinton administration and continuing through the Obama administration.
As the population grew during the Clinton years, more babies were probably born under the Bush administration than the Clinton administration, due to a larger population, and as Obama has not, as yet, served the entire 8 years which Bush served, I would suspect more babies (by a very small difference) were born during the Bush administration than the other two, so far. Next year at this time the honor will probably go to Obama.