Not to get political, but it's often used online to criticize fiat currency and entitlement spending. Surely he couldn't have imagined fiat currency or things like Medicare, so what was the issue in his day?
Its almost certainly a misattribution, something that happens with the founders all the time. The earliest source I could find for that quote is an article in the Daily Oklahoman from Sunday December 9th, 1951 The quote there is
“A democracy cannot exist as a permanent form of government. It can only exist until the majority discovers it can vote itself largess out of the public treasury. After that, the majority always votes for the candidate promising the most benefits with the result the democracy collapses because of the loose fiscal policy ensuing, always to be followed by a dictatorship, then a monarchy."
While not exactly the text you sited, the similarity is clear. However, the author attributes it to Alexander Fraser Tytler (15 October 1747 – 5 January 1813) a "obscure Scotsman." While I could not find an original source in Tytler's writings, it is in line with his somewhat negative feelings about Democracy. So, it may or may not originally be a Tytler quote, but I can say with near certainty that is it not a Franklin quote.