There is no such thing as bad publicity
The idea that no publicity can do harm is clearly open to question. For someone seeking notoriety and a somewhat scandalous reputation, like Marie Lloyd and Mae West in days gone by, or Paris Hilton in our era, that may be true. The shareholders of BP and Toyota, which have both suffered falls in their market price due to worldwide publicizing of their recent difficulties, may feel differently.
'There's no such thing as bad publicity' is often associated with Phineas T. Barnum, the 19th century American showman and circus owner. Barnum was a self-publicist of the first order and never missed an opportunity to present his wares to the public. As with many other supposed quotations, there's no hard evidence to link the 'bad publicity' quotation to him.
The proverbial expression began to be used in the early 20th century. The earliest version that I have found in print is from the US newspaper The Atlanta Constitution, January, 1915:
All publicity is good if it is intelligent.
The thought behind the proverb had been expressed earlier by Oscar Wilde:
The only thing worse than being talked about is not being talked about.
Probably the most celebrated adapter of the expression was another great wit from the Dublin literary scene, the Irish Republican and "drinker with a writing problem", Brendan Behan. Behan's boisterous lifestyle meant that for him, more than others, there was truth in his opinion that:
There's no such thing as bad publicity except your own obituary.
See also: the List of Proverbs.