Never; unless there is a specific law in the jurisdiction that requires it, however, that is a separate crime, it doesn't make you an accomplice to the first crime.
There is no general obligation to report a crime; some jurisdictions may have legislated to make reporting mandatory, either generally or for specific professions.
From the link, an accomplice is:
One who knowingly, voluntarily, and with common intent unites with the principal offender in the commission of a crime. One who is in some way concerned or associated in commission of crime; partaker of guilt; one who aids or assists, or is an Accessory. One who is guilty of complicity in crime charged, either by being present and aiding or abetting in it, or having advised and encouraged it, though absent from place when it was committed, though mere presence, Acquiescence, or silence, in the absence of a duty to act, is not enough, no matter how reprehensible it may be, to constitute one an accomplice. One is liable as an accomplice to the crime of another if he or she gave assistance or encouragement or failed to perform a legal duty to prevent it with the intent thereby to promote or facilitate commission of the crime.
Many jurisdictions impose an obligation on certain professions to report suspected crimes. The most common and obvious being a police officer who generally has an obligation to report all suspected crimes. Other professions include doctors, teachers, nurses etc. in the case of suspected child abuse; sometimes this extends to elder abuse but usually doesn't include an obligation to report spousal abuse.
If a person with such an obligation fails to report, they have not become an accomplice to the original crime or any future crime; they have broken a different law on their own.
In general, a witness to a crime is not required to report it. They could be asked to give a statement but are not obliged to and would not be subject to arrest. If they were subpoenaed to appear in court as a witness then they would be obliged to do so and give evidence; failure to do either would be contempt of court.
To be clear, a perpetrator of a crime is also not obliged to report it and is protected from contempt by rules about self-incrimination like the US Fifth Amendment. As an aside, NSW, Australia has recently passed a law that while maintaining the accused's right not to testify has allowed juries and judges to draw inference from the silence in their deliberations.
Some jurisdictions have specific laws that make it a crime not to report. For example, in New South Wales, Australia Section 316 of the Crimes Act makes it an offence to conceal a serious indictable offence (i.e. one with a maximum penalty or 5 or more years gaol) without "reasonable excuse". The punishment is up to 2 years; if you solicit or accept any benefit for concealing it, the punishment is 5 years. Prosecution for people who came by the information as a result of practicing certain professions requires permission of the Attorney General.
- The bar fight witness has no obligation to report and is not an accomplice. In NSW, this is the crime of affray, a serious indictable offence, and must be reported.
- The mother of the child has no obligation to report and, unless she is assisting or encouraging the commission of the sexual assault, is not an accomplice. Further, spousal testimonial privilege does not apply; i.e. the mother could be compelled to give evidence against the boyfriend or face contempt. In NSW, this is the crime of Sexual intercourse-child under 10, a serious indictable offence, and must be reported.
- The passenger has no obligation to report and is not an accomplice. In NSW, this is the crime of failing to stop and assist after vehicle impact causing death or grievous bodily harm, a serious indictable offence, and must be reported.
At what point does one's involvement become such that they are guilty simply for failing to report another individual, assuming they have not actively done anything to directly support the crime otherwise?
At the point where the person moves from being a witness to a participant by assisting or encouraging the perpetrator.
... if someone is indirectly benefiting from a crime they do not encourage or facilitate, ..., does this make them an accomplice?
No, however, if the person knows that the benefit are the proceeds of crime then they could be charged with receiving stolen property and would not have good title in the property even if they didn't know.
presumably a bar serving a drink to someone they know is a bookie doesn't make them guilty, even if the bookie presumably earned the money he is using to buy the drink via an illegal job?
It depends if they know the money is the proceeds of crime; if they do then receiving it is a crime in its own right but it doesn't make them an accomplice.