The advice came after the force used Facebook to appeal for information about Jermaine Taylor from Newport, who breached his licence conditions after being released from prison in December.
The 21-year-old's mugshot, which was posted alongside the online appeal, shows the man with a receding hairline - which has been the subject of jokes and puns in the comment section.
In an effort to stop people mocking the man, Gwent Police warned Facebook users: "Please remember that harassing, threatening and abusing people on social media can be against the law.
"Our advice is to be as careful on social media as you would in any other form of communication.
"If you say something about someone which is grossly offensive or is of an indecent, obscene or menacing character, then you could be investigated by the police."
The photo of Taylor has been liked more than 9,000 times, commented on 80,000 times and shared 13,000 times.
Taylor originally received a three-year sentence for being concerned in supplying cocaine after being sentenced at Cardiff Crown Court on 4 September 2017.
He has been recalled to prison for breaching his licence release conditions.
Gwent Police did not respond to a request to comment.
:: If you have any information on Jermaine Taylor's whereabouts, please call 101 quoting 1900283943 or Crimestoppers on 0800 555 111.