Should "The history of X" be followed by "began" or "begins"?
This BBC page states
The history of life on Earth began about 3.8 billion years ago...
Similarly, this one states
The history of Canterbury's religious significance began with the Romans...
However, this page states
The history of Rastafari begins with the colonisation of Africa...
as does this:
The history of the march begins in the village of Tolpuddle...
Which is correct/preferred, or are both equally acceptable?
This is a stylistic choice, called the narrative time. In the first two examples, the narrative past is used; it is by far the most common narrative style of recounting history or stories (once upon a time, there lived...).
In the second, the narrative present (also called the historical present) is used. It is as an attempt to draw the viewer/reader into that particular story's time-frame as if they are there to witness it as it is happening. Interestingly this is more common is spontaneous speech (recounting of facts or story) than in writing (...and then he says to her, ...).
The choice is yours.