Nicola Sturgeon has warned so-called "cybernats" who post abuse online that they could be disciplined by the party.
The Scottish First Minister said it is "not acceptable" for people to use social media to "threaten violence, or hurl vile abuse, or seek to silence the voice of others through intimidation".
If SNP members "cross the line" in their posts, she said the party will take action.
Writing in the Daily Mail, Ms Sturgeon pledged: "When tweets or postings from SNP members that cross the line are brought to our attention, we will act - as we have done before.
"That is why I am making clear today that the SNP will take steps to warn those whose behaviour falls short of the standards we expect.
"We will tell them to raise their standards of debate, to stick to issues not personalities, and to ensure robust and passionate debate takes precedence over abuse and intemperate language.
"I am also making clear that where appropriate we will take disciplinary action. In the SNP we have a code of conduct and online guidance for our members.
"Where that code is broken, members should have no doubt that we will use our disciplinary processes."
Ms Sturgeon, who has some 229,000 followers on Twitter, said the level of abuse she receives on social media "would make people's hair stand on end were they to see it".
She added: "I choose simply to ignore it, but that doesn't mean that online comments which cross the line of decency are acceptable.
"Where political disagreement is passionate, open, honest and robust and conducted with respect, it is welcome. Even where views are expressed using language that I wouldn't use, I accept that - after all that's in the nature of free speech.
"But where people use Twitter to threaten violence, or hurl vile abuse, or seek to silence the voice of others through intimidation, that is not acceptable - and we must all say so loudly and clearly."
She also urged other political leaders to follow her example, saying: "We must send out a clear message that politics in Scotland will not be sullied by this behaviour.
"Raising the standard of debate is a responsibility across the board and I urge all parties to do as we have done - to say clearly that crossing the line will not be tolerated."