The answer should be roughly 1/365 of the Earth's population (more on that below). The current world population estimate from the US Census Bureau can be found at:

http://www.census.gov/ipc/www/popclockworld.html

As of March 1, 2009 they estimate the world population to be ~6,763,611,245 (which we'll call about 6.7 billion). If we assume that births are randomly distributed throughout the year (which isn't quite true), there are therefore about 18.4 million people around the world sharing a birthday each and every day.

As to the question of the distribution of births throughout the year, the number of average births does indeed vary somewhat as a function of the calendar month. According to:

http://www.sciencenews.org/sn_arc98/11_21_98/mathland.htm

"In the United States, the data show a seasonal pattern, varying between 5 percent above and 7 percent below the average daily frequency."

Five percent of 18.4 million is about 1 million people. So we can approximately claim that on any given day 18.4 +/- 1 million people are likely to share that date of birth.

Finally, you may want to check out the Wikipedia entry for "Birthday problem" (and references therein) which describes the mathematical process for calculating the probability that a common birthday will exist in a set of randomly chosen people.
Source(s):
http://www.sciencenews.org/sn_arc98/11_21_98/mathland.htm
http://en.wikipedia.org/wiki/Birthday_paradox


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