The EEOC has authority to bring civil claims against companies in cases of sex discrimination, of which sexual harassment is a sub-type. Indeed, generally speaking, you can't bring a federal sex discrimination lawsuit against a private company without first presenting it to the EEOC.
But, to the extent that the conduct doesn't exclusively involve employees or prospective employees, the EEOC would not be involved in those matters, and the Civil Rights Division of the Justice Department and private attorneys for the victims might have concurrent jurisdiction to bring civil claims.
Can federal prosecutors investigate a pattern of sexual harassment
inside a company
When the EEOC gets a complaint, it can expand the scope of its case beyond the person bringing the complaint. Saying "federal prosecutors" in this context is somewhat misleading, however, since that term is usually reserved for federal government lawyers who bring criminal cases, rather than civil sexual harassment charges.
I don't recall whether the EEOC brings civil cases discrimination cases that goes to trial with its own lawyers (I think that this is usually the case) or with Justice Department lawyers. Both, in any case, are federal government lawyers, and in neither case would this be a criminal prosecution.
The EEOC can choose not to bring the claim, in which case a private attorney for the victim can bring it.
Sexual harassment, in a narrow sense, is not a state or federal crime, per se, although often sexual harassment involves conduct which is a state and/or federal crime. Usually, state law has a crime for unwanted sexual contact, a crime for unwanted sexual intercourse, a crime for extortion, and a crime for coerced prostitution, to name a few. I won't attempt to enumerate the potentially relevant federal crimes which usually must have some form of "federal" hook (e.g. transporting someone across state lines for purposes of sexual activity, or raping a federal employee).
State crimes can be prosecuted by the state prosecutor (often both a county or district attorney and a state attorney general with concurrent jurisdiction, but sometimes integrated into a single agency with local branches in others like Florida).
Federal crimes can be prosecuted by a federal prosecutor, usually a U.S. Attorney's office for a U.S. District Court District (which is part of the Justice Department), but sometimes federal crimes are prosecuted by lawyers assigned to a particular national level sub-division of the Justice Department (e.g. the Civil Rights Division). Private parties cannot prosecute crimes as a general rule (with some exceptions in a handful of cases in a few states most in the East).
It wouldn't be unheard of to have one set of federal government lawyers press a civil EEOC claim for sexual harassment, while a separate team of federal government lawyers press criminal charges related to the same conduct, with the separate teams of lawyers coordinating with each other to some extent.
It would be exceedingly rare to have the same lawyers handling both civil and criminal charges at the same time in a sexual harassment fact pattern. There would almost always be different lawyers handling civil cases and criminal cases in the federal government.
Can the acts of executives and assistants who aided Weinstein be
viewed as solicitation?
Individuals who personally participate in some manner in the civil tort of sexual harassment or the crimes charges could have civil or criminal liability respectively in most circumstances.
UPDATE: The FBI is basically investigating this kind of thing right now per Fox News. But, the FBI is an investigative/law enforcement agency, rather than attorneys prosecuting a case (even though a large share of FBI agents have law degrees). Federal agencies have broader powers to investigate potential crimes or civil rights law violations than they do to prosecute claims. Investigation requires only probable cause to believe that there was wrong doing, or even a reasonable suspicion or less for many forms of investigation (like asking for voluntary testimony from witnesses without conducting any 4th Amendment searches), even if there is not more solid evidence of it at the outset.
Federal agencies also often have statutory civil subpoena power to investigate if there have been violations of particular federal statutes that does not require a showing of probable cause and can be utilized prior to bringing a civil action or criminal charges against anyone.