A blend of banker and gangster, the noun bankster denotes a member of the banking industry seen as profiteering or dishonest.

It is distinct from bankster in the neutral sense of a bank employee, which is composed of
– the noun bank
and
– the suffix -ster, denoting a person engaged in, or associated with, a particular activity—as in the nouns songster and gangster for example.

The earliest occurrence that I have found of bankster in this neutral sense is from the Buffalo Morning Express (Buffalo, New York) of 20^th January 1889:

Mr. William C. Cornwell has published a table showing that banks in New-York State have 113 legal holidays in the year 1889!
This puts the average bank employee pretty nearly on a par with peasants in Italy, who have so many saints’ days to observe that a day’s work becomes a great treat to them (?)
The unfortunate bankster can appreciate the reply of the hulking tramp who complained, when asked why he didn’t go to work:
“I can’t work. I’m too strong.”

The earliest instance that I have found of bankster as a blend of banker and gangster is from The Muncie Morning Star (Muncie, Indiana) of 22^nd December 1931:

Today
A New Word, Banksters.
[…]
By Arthur Brisbane

Cyrus H. K. Carter receives from Paris information that American bankers are called “banksters,” in high French financial circles.
A “bankster” is supposed to be a combination of banker and gangster, expressing French opinion of American bankers.
New York bankers may reply that in dealing with the more clever of the distinguished French bankers a little of the gangster quality would not come amiss, for purposes of self defense.

According to that newspaper article therefore, bankster as a blend of banker and gangster first appeared in French in 1931.

In that case, it must have been a spoken use only, because, according to the documents currently available on gallica.bnf.fr / Bibliothèque Nationale de France, the word did not appear in print until 1933, and as a borrowing from American English.

The earliest occurrence that I have found is from an interview of General Marchand, President-General of the Colonial War Veterans, published in L’Éveil de l’A.E.F. (The Awakening of F.E.A. – Brazzaville, French Equatorial Africa) of 4^th March 1933:

« Nous ne voulons pas, continue le général, que la France, jusqu’en ses terres les plus lointaines, devienne la proie des gangsters, ni, comme on dit assez drôlement à Wall Street, des Banksters. »
     translation:
“We do not want, the general continues, France, as far as its most distant lands, to become prey to the gangsters, nor, as it is rather funnily said at Wall Street, to the Banksters.”

The second-earliest instance of the word in French is from a correspondence from Washington, D.C., published in Paris-Soir (Paris) on 21^st March 1933; the correspondent, Jules Sauerwein, quoted a high-ranking federal civil servant as saying:

« En ce moment, le bouc émissaire est Wall Street en particulier et les banques en général. On appelle les banquiers des banksters, mot qui rime avec gangsters. »
     translation:
“At the moment, the scapegoat is Wall Street in particular and the banks in general. The bankers are called banksters, a word rhyming with gangsters.”

Another early occurrence is from an article titled « Banksters », published in Le Matin (Paris) of 10^th May 1933, and written from New York City by the French journalist and novelist Joseph Kessel (1898-1979):

Le terme de gangster a été si fortement répandu en France par la presse, le livre et le film, qu’il a presque reçu ses papiers de naturalisation. Aussi le jeu de mots qui sert de titre à cet article et qui, de l’autre côté de l’Atlantique, désigne couramment les banquiers et Wall Street, est-il aussi compréhensible, aussi valable et typique à Paris qu’à New York.
Gangster, — Bankster !
Ce rapprochement en dit long sur l’état d’esprit du peuple américain à l’égard de ses financiers. Il n’est pas né sous la plume d’un humoriste, d’un satirique, d’un pamphlétaire. Il ne provient pas d’une trouvaille heureuse d’un orateur indigné, inspiré. Il s’est formé d’une manière anonyme, obscure et spontanée, dans la masse et, un jour, a éclaté sur toutes les lèvres, comme une formule lapidaire, cruelle, répondant exactement au sentiment général.
     translation:
The term gangster has been so widely spread in France by the press, the book industry and the film industry, that it has almost received its naturalisation papers. Therefore, the pun which serves as the title of this article and which, on the other side of the Atlantic, commonly designates the bankers and Wall Street, is as comprehensible, as valid and typical in Paris as in New York.
Gangster, — Bankster!
That parallel speaks volumes about the state of mind of the American people towards their financiers. It did not come into existence under the pen of a humorist, of a satirist, of a pamphleteer. It is not due to a brainwave of an indignant, inspired orator. It came into being in an anonymous, obscure and spontaneous fashion, in the masses and, one day, burst out on everyone’s lips, as a lapidary, cruel formula, answering exactly the general sentiment.