Is it the Chinese government responsibility to monitor and punish Chinese fishing ships illegally entering a country EEZ to fish illegally?
No. As far as I'm aware China has made no commitment to counter illegal fishing by its trawlers outside of China.
Can China do anything to prevent this, or is it considered to be outside of China's jurisdiction and to be the responsibility of the victimized countries to monitor and punish these offenders?
Yes, China can do something about illegal fishing. The main reason for that is that China is a port state (it has many ports) and that illegal fishers need to unload their catch somewhere before it can be sold. When many ports work together to refuse to harbor ships engaging in illegal fishing, then that's a big hurdle to overcome as an illegal fisher.
One important treaty to counter illegal fishing is the Port State Measures Agreement (PSMA). The primary goal is to regulate fishing vessels docking in member states and to prevent them from unloading fish that was caught illegally. According to Wikipedia:
The treaty requires that fishing vessels request permission to dock at a port and inform the port of the details of its fishing operations. Permission to dock can be denied if unregulated fishing occurred. The measure is intended to block illegally caught fish from entering the marketplace. Other measures in the treaty include inspections of equipment, paperwork, catches, and ship's records. Though the treaty does not compel countries to apply these measures to ships under their own flags, they may choose to do so under the agreement.
If China would sign up to PSMA, it would have to investigate when another PSMA member state refuses entry to a fishing vessel flying the Chinese flag if they suspect the ship of having engaged in illegal fishing. The results of the investigation would have to be shared with the FAO. According to an article on maritime-executive.com:
The PSMA, which is administered by the UN Food and Agriculture Organisation (FAO), mandates that countries require vessels that fly their flag to submit to port inspections. If a port state denies a ship entry because of suspected illegal fishing, it must notify the flag state – the country where the vessel is registered. If the flag state is a party to the PSMA, that country then has the duty to investigate the vessel and, if it finds evidence of illicit fishing, take action to penalise the ship, such as by levying fines or revoking its registration.
The flag state must then report the result of that investigation and the actions it has taken to the FAO, relevant port states and regional fisheries management organisations.
In summary, if China wanted to counter illegal fishing as described in your question, then it could do so by signing up to and strictly enforcing PSMA (even on ships flying the Chinese flag). That way, Chinese ships would be subject to investigation if there is a suspicion of illegal fishing when they dock in a country that is party to PSMA (including China).