The answer to this is, it depends on a variety of factors, including what type of crime it was, the severity of the punishment and the age you were when the offense was committed.
Commission of certain criminal offenses, known as Crimes Involving Moral Turpitude (CIMTs) can render the applicant inadmissible to or excludable from the United States.
These offenses generally speaking incorporate aggravating factors such as extreme violence, or depraved or dishonest acts on the part of the perpetrator.
Examples of such offenses are rape, aggravated assault, homicide and larceny.
Other offenses are less serious and therefore are not regarded under U.S. law as CIMTs. 
Even if you have committed a CIMT, you may fall under one of the exceptions such as the Petty Offense exception and the Juvenile exception.
The U.S. Embassy, London’s stance is unequivocal and it states that if you have ever been arrested or convicted for any criminal offense, you must apply for a visa and inform them of the circumstances surrounding the commission of the offense/s.
Hence if you have ever been arrested and/or convicted for the commission of a criminal offense, no matter how minor, please contact us immediately so that we can advise of the immigration implications of that offense and in order to advise what to do next.
Please Note: Travel to the United States after having committed CIMTs, without following the proper procedure can have severe adverse consequences.