The former Cabinet minister said she received the message via her Twitter account on Tuesday.
The comment accused her of sending pro EU "propaganda", adding "Please stop or I will kill your kids and grandkids".
The incident comes amid increased security for MPs following the death of Jo Cox.
Mrs Cox died after being shot and stabbed in an attack in Birstall near Leeds on Thursday. A 52-year-old man has been charged with her murder.
Ms Cooper, the Labour MP for Normanton, Pontefract and Castleford, reposted the message on her Twitter page and added: "Got this today for speaking out for Remain... This has to stop."
The account from which the message was sent now appears to have been removed.
West Yorkshire Police said they had attended the MP's office and taken statements.  The force added:  "A crime of a malicious communication has been recorded and is being investigated today by Wakefield detectives."
Ms Cooper was recently involved in a campaign to make the internet less aggressive, sexist, racist and homophobic.
In the wake of the death of Mrs Cox, MPs have been asked by the police to review security at constituency surgeries.
Labour's Neil Coyle said MPs had been "warned of copycat attacks", while Ben Bradshaw said they had been told to hold appointment-only surgeries.
The Conservative MP for South Thanet, Craig Mackinlay, contacted Kent police after images of a gun and a knife and the message "Another MP that needs..." were posted on his Facebook page.
Mr Mackinlay said: "Whilst I can take the rough and tumble of modern politics, my wife and staff are obviously concerned about their daily activities in serving the public."
MPs have previously been victims of abuse on social media.
In 2014, a man with links to an extremist right-wing group was jailed for four weeks for sending an anti-Semitic tweet to Liverpool Labour MP Luciana Berger.
The Labour MP for Walthamstow Stella Creasy had a panic button installed at her home after she was targeted online by a man who threatened to rape her for supporting a campaign for a female figure to appear on a Bank of England note.
Peter Nunn, 33, from Bristol was jailed for 18 weeks and issued with a restraining order in September 2014 after he was found guilty of sending indecent, obscene or menacing messages to Ms Creasy.
Labour MP Yvette Cooper has contacted the police after receiving a message she said threatened the lives of her children.