The owner of the trademark, not the web site, is responsible for policing counterfeits. It is a complex area of law still being litigated (with different results) around the world.
The above is general legal and business analysis. It is not "legal advice" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
YES, unless they receive a proper DMCA take down notice. They are likely "not doing anything" [which probably means to you not removing a specific listing from among the millions posted on their site] because they have not received such a notice or because the actual people running Craigslist don't even know about the particular listing. Craigslist is largely automated, so "they" is likely a computer that has no way know of telling if something is real or counterfeit. It is up to the brand owner to police their brand. And, how do you even know what Craigslist is or isn't doing behind the scenes or on other fakes? My guess is they remove anything they are officially notified is counterfeit or illegal. If counterfeits are causing you trouble, see an Internet attorney, who will know how to properly notify them.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
Find a human, make oral and written demand that they de-list the goods. Craigslist doesn't need the hassle.
Professional Rules of Ethics require me to advise you that this is an offer of possible representation.