Here is an interesting explanation offered by Robert Hendrickson, The Facts on File Dictionary of Word and Phrase Origins (1997):

racket; racketeer. English pickpockets, once the best of the breed, invented the ploy of creating disturbances in the street to distract their victims while they emptied their pockets. This practice was so common that a law was passed in 1697 forbidding the throwing of firecrackers and other devices causing a racket on the city streets. From the common pickpocket ploy the old onomatopoeic English word racket, imitative like crack or bang and meaning a disturbance or loud noise, took on its additional meaning of a scheme, a dodge, illicit criminal activity. Before 1810, when it first appeared in print, the word had acquired this slang meaning in England, though it was later forgotten and the word racket for a criminal activity wasn't used again there until it was reintroduced from America along with the American Prohibition invention from it, racketeer. The only other, improbable, explanation given for the word is that it was originally the name of an ancient, crooked dice game. ...

Eric Partridge, A Dictionary of Slang and Unconventional English, Fifth Edition (1961) corroborates Hendrickson's etymological analysis:

racket. A dodge, trick; plan; 'line', occupation, esp. if these are criminal or 'shady': ... Ex. racket, noise, disturbance. ...

John Hotten, A Dictionary of Modern Slang, Cant and Vulgar Words (1859) has this entry:

RACKET, a dodge, manœuver, exhibition ; a disturbance.

Francis Grose & Pierce Egan, Grose's Classical Dictionary of the Vulgar Tongue (1823) has this:

RACKET. Some particular kinds of fraud and robbery are so termed, when called by their flash [that is, cant] titles, and others, Rig ; as, the Letter-racket ; the Order-racket ; the Kid-rig ; the Cat & Kitten rig, &c. but all these terms depend on the fancy of the speaker. In fact, any game may be termed a rig, racket, suit, slum, &c. by prefixing thereto the particular branch of depredation or fraud in question, many examples of which occur in this work.

So the noun racket in its underworld sense has been around since the early 1800s (at least) and appears to have been inspired by the use of sudden noises by pickpockets or their confederates as a distraction just prior to a theft.

--Sven Yargs

Shady business practice.