Well, not exactly in the '60's, like people think. From StraightDope:
"If you thought the term pig arose in the 1960s, you're in for a surprise. The OED cites an 1811 reference to a "pig" as a Bow Street Runner--the early police force, named after the location of their headquarters, before Sir Robert Peel and the Metropolitan Police Force. The usage was probably confined to the criminal classes until the 1960s, when it was taken up by protestors. False explanations for the term involve the gas masks worn by the riot police in that era, or the pigs in charge of George Orwell's Animal Farm."
From Groink, we have another explanation, nearly as old, and also referring to Sir Robert Peel and his men, with a bit clearer of an explanation as to why they were pigs:
"Police - pigs
Back in 1809, Sir Robert Peel entered the House of Commons in London - he developed a passion for Sandy Back pigs found in Ireland and began to breed them in Tamworth. Soon, these pigs were known as Tamworth pigs. Pig slang was commonplace in Tamworth because of this, it was in 1829 that the relation to police came into it.
Politicians were concerned about the way London was policed and Sir Robert Peel changed things - his changes resulted in the formation of the Metropolitan Police. This is why police are referred to as 'Bobbies' or 'Peelers'; they were Bobby's boys...
Due to the pig nature that Tamworth had become, the police suffered the same fate as other Tamworth products did: They became related to pigs. "
So, it's thanks to the Brits, for this bit of slang, it seems.