Krishna's historical situation appears to be similar to that of Jesus: there's probably enough consistent historical evidence to say that he existed at one time as a human being, and a few other details of his adult life are probably very roughly historical, but that's about it. Birth narratives (in both cases) are particularly inconsistent, and untrustworthy.
Here's what Historian Guy L. Beck wrote on the subject:
The empirical evidence of inscriptions, dated monuments, and original manuscripts is not perhaps as strong for Krishna as in some of the other examples of religious figures. However, most scholars of Hinduism and Indian history accept the historicity of Krishna - that he was a real male person, whether human or divine, who lived on Indian soil by at least 1000 BCE and interacted with many other historical persons within the cycles of the epic and puranic literatures.
Note that the birth narratives of Krishna, Jesus, Moses, and Zeus all share the plot element that the ruler wanted to kill the baby due to a prophecy. This is a common theme in mythology literature, and thus kind of a dead giveaway that the account was written with goals other than historical accuracy in mind*
* - ...or more bluntly, it was largely invented.
Meta question, brought up in chat: Why can't there be more information in this answer?
Lets take my other example of Jesus. He was born about 3,000 years later, into a thriving literary culture (The Roman Empire). Additionally, traditionally Christian areas happen to be the region of the world that contains the most research institutions, so there's always been plenty of motive and opportunity for historians to study his life.
With all those advantages, secular historiography on the subject can come up with only 6 generally agreed facts about his life (none of which have anything to do with his birth). That's it.
The stories about Krishna are set roughly around 3,000BCE, and thus would have been 2,000 years old by the time they were even first written down. Time-wise, that's like if we were just now bothering to first write down the various stories circulating about Jesus.
Not that I'm implying they were lazy, they just didn't have writing yet.
There was an Indus Valley script in use in NW India at that time, but it is undeciphered, was likely used by a separate people, and was a 400+ symbol logographic system, which would have drastically reduced the amount of people capable of using it.
So it really shouldn't shock anyone that we just can't say as much about Krishna