The Lord Advocate issued the direction after police launched an investigation into posts made about the tragedy, which killed nine people when a police helicopter crashed into the Clutha bar on November 29.
It is understood about one dozen cases are being examined, some involving offensive, racist or sectarian comments about the Glasgow crash.
A 16-year-old boy was arrested last week for allegedly posting sectarian and racist comments online about the incident.
Lord Advocate Frank Mulholland wants a "robust prosecution policy" and guidance is to be issued to prosecutors this week on how they should deal with hate crimes linked to the Clutha helicopter tragedy.
Procurators fiscal will be told that, where it can be demonstrated an offence was motivated by a reaction to events at the Clutha, there will be a presumption in favour of criminal proceedings.
The Lord Advocate said: "It is important that the Crown Office and Procurator Fiscal Service demonstrates a robust prosecution policy towards such offences committed as a reaction to the incident in recognition of the fact that people died and the impact such crimes will have on their families and friends.
"This is also in consideration of the other people who were in the Clutha bar and those who attended the scene in the aftermath.
"I have made it clear that prosecutors must take a hard line against this kind of hate crime."
The incidents are some of the most prominent to come under the 2012 Offensive Behaviour at Football and Threatening Communications (Scotland) Act.
David McKie, a lawyer with Glasgow-based solicitors Levy and McRae, said the message from the Lord Advocate to the public was one of zero tolerance.
He said: "The Lord Advocate is no doubt intending this statement not just to cover internet communications on this tragedy, but also future conduct in general.
"If there are clear examples of sectarian statements being made in cases of this type then the Lord Advocate's position indicates a greater likelihood of prosecution.
"However, the law is still relatively untested and, where statements are made that do not have obvious sectarian content, or where any perceived threat to the public is debatable, then the outcome remains less certain."
There have been several recent cases where people have been prosecuted for inappropriate messages posted on social media sites such as Facebook and Twitter.
The action followed the introduction of the new laws to tackle bigotry and hatred in football.
The Act created new offences relating to behaviour deemed to incite religious, racial or other hatred specifically linked to football matches. It also targets online abuse with a sectarian element.
Last year, a 20-year-old man was arrested for threats aimed at Celtic manager Neil Lennon on a social network. A Twitter troll who abused the memory of Celtic skipper Scott Brown's dead sister was also prosecuted last year.
lA man has been convicted of a religiously aggravated breach of the peace after subjecting a journalist who edited a book on the financial demise of Rangers to a torrent of abuse on an internet broadcast. David Limond, 41, targeted Angela Haggerty on his Rangers Chat show. Sheriff Scott Pattison convicted Limond At Ayr Sheriff Court.
Sentence was deferred for reports. The sheriff said: "I view this as very serious and I am strongly considering a custodial sentence."