Which is correct and why,"The officer is accusing the individual of having committed a crime" or "The officer is accusing the individual of committing a crime"?
These are both essentially correct, however there is one small difference:
(1) Accusing someone of having committed a crime can only refer to the past. That is, the policeman is referring to a crime that has already (allegedly) been committed:
(2) Accusing someone of committing a crime, can refer to an act that is in progress at the time of the accusation (present):
Note however, that (2) can be used to refer to the situation in (1) if the context is clear.