Here are a couple of instances of "Frankenstein" in the sense of "Frankenstein's monster" from a few years before those that Laurel cites in her very useful answer. First, from "English News: House of Commons," in the Sydney [New South Wales] Monitor (August 20, 1834):
We direct attention to several articles of news which appear in our paper of to-day, relating to the proceedings of Trades' Unions, and the revolution in the state of society which they are tending to accomplish.
These Unions seem likely to turn out the real Frankensteins of the Whig Ministry. Whig agitation gave them birth when the Whigs were in opposition ; but the patronage and countenance of Government, it was, which imparted to them vigour, and a sense of power. The Government taught them that they had power, and encouraged them to use it against the supremacy of the law.
And from a letter to the editor of the [Hobart, Tasmania] Colonial Times (August 2, 1836):
It is also too true that Umbra [a letter writer to a rival newspaper] possesses persuasive eloquence, which can make "the worse appear the better reason;" nay, he has more—much more ability of head—but sad to say his heart is wanting wholly; like Frankenstein's he's incomplete, though like that monster he has his victims.
Even earlier is this OCR snippet from an article in the London [England] Globe (August 22, 1826):
Mr. T. P. Cook expected to return [to] London immediately, in order appear the English Opera-house in the same character which has been much the rage at Paris, the monster Frankenstein.
Instances of popular association of "Frankenstein's monster" with "the monster Frankenstein" thus seem to date back to the middle 1820s at least. The process may have been helped along by popular dramatic entertainments in which Mary Shelley's title, Frankenstein, or The Modern Prometheus was recast first (in 1823) as The Fate of Frankenstein and later (in 1827) as Frankenstein, or The Monster.