This is a very complex question with different answers depending on the type of content we are talking about, the country at issue, and whether we are talking about legal or ethical responsibilities.
For example, as you can see in one of the links below, the European Court of Justice said that ISPs cannot be forced to monitor content for the purpose of reducing internet piracy. This means that in that jurisdicition, at least, there is no legal responsibility to monitor content. The same is generally true in the United States. On the other hand, we can also see that the government of India wants Facebook and Google to monitor and block access to objectionable material.
Having seen that legal responsibility varies, is there an ethical responsibility? Here, I would argue that there is not. It is too much to ask of an ISP that it spend its resources on trying to monitor all the content on all websites that use its servers. The responsibility is with the people who make the websites. We do not ask the owners of office buildings to investigate their tenants' business dealings. Neither should we ask ISPs to investigate what their users are doing.