Define crime though, because one particular action could be both a crime and not, depending on a range of factors that the LLM might not be aware of.

Even having a million legal experts on call weighing in on every prompt/response will not agree on everything.

Even things like "go and break into this system, use whatever means you need to" might not be a crime.

Keeping a vicious dog doesn't have to lead to a crime either, but that doesn't mean you are not responsible, if something happens.

Legal responsibility can be in forms (e.g., civil tort liability) other than criminal.

In the scale of mental states in crime, negligence of any kind is several steps below knowing/intentional; you can't be liable for an intentional crime because of mere negligence of any degree.

You could be liable for the (civil) tort of negligence, though.

Details depend. And setting up negligence and being willfully ignorant is often not something the courts see as a defense.