And anyone who sent "hate" mail could be sentenced to six months in prison or fined £5,000 - double the amount at present.
Home Secretary Jack Straw has drawn up proposals to update the law to make it an imprisonable offence to send hate mail by traditional or electronic means.
Mr Straw is also planning to give police greater powers to stop protestors from gathering outside people's homes.
Amendments
The changes will be contained in amendments to the Criminal Justice and Police Bill now going through parliament.
They will be tabled in the Commons on Thursday.
Under the plans, protestors who refused a police request to move away from a private residence could be arrested and face up to three months imprisonment. They could also be fined up to £2,500.
And anyone who sent "hate" mail could be sentenced to six months in prison or fined £5,000 - double the amount at present.
Mr Straw is also planning to make it harder for protestors to defend their actions.
An amendment to the Malicious Communications Act would establish an "objective test" to be used to decide if their actions were reasonable.
At present it is a defence under the Act for the sender of hate mail to say that they honestly believed their threats were reasonable.
Under the amendment, a court would be able to convict a protestor if it believed that a reasonable person would not have taken the same action.
Violence and intimidation
Mr Straw said: "We simply will not tolerate the criminal actions of a small number of extremists who use violence and intimidation to stop people going about their legitimate business.
"These new measures are designed to help prevent two tactics often used by these individuals - protesting outside employees' and directors' homes and sending intimidating mail.
"We want to ensure that all types of threatening messages are covered - including those sent by text messaging and email.
"Tougher penalties for sending hate mail will be an added deterrent."
Guidelines
The Home Office is also working on guidelines for shareholders and employees of research companies who may be targeted by animal rights extremists.
The guidelines will outline steps they can take to improve their personal security and avoid unnecessary publication of their home addresses.
Mr Straw said: "We are determined to make sure that the public are protected and that lawful businesses can operate freely."
He added: Peaceful public protest is entirely legitimate, but there is a world of difference between that and the type of intimidation, threats and violence we have seen recently."