One BJP MP, Pratap Simha of Karnataka, compared Gurmehar Kaur to wanted gangster Dawood Ibrahim, while Union minister Kiren Rijiju suggested her mind was "polluted". One person, on Facebook, threatened to rape her.
"You may not like what the person thinks but...Mindless threats, trolling emanating from people whom the Prime Minister of India follows (sic). This is the worst form of state intimidation and this is not how democracies and democratic countries function," said Congress spokesman Manish Tiwari.
In her campaign, she has blasted the ABVP, the RSS-aligned students wing, for instigating violence at New Delhi's Ramjas College last week.
"The brutal attack on innocent students by ABVP is very disturbing and should be stopped. It was not an attack on protesters, but an attack on every notion of democracy that is held dear in every Indian's heart.
"It is an attack on ideals, morals, freedom and rights of every person born to this nation," Kaur said in a Facebook status.
To mock her, Simha also posted a photograph showing Dawood with the message, "I didn't kill people in 1993. Bombs killed them."
Rijiju also took to Twitter to take a dig at her.
"Who's polluting this young girl's mind? Strong armed forces prevent a war. India never attacked anyone but a weak India was always invaded," the minister of state for Home Affairs said.
Apart from the Congress, the AAP and the CPI(M) also
reacted sharply and demanded "strict action" against those who allegedly threaten to rape Kaur.
"Women are afraid to speak their mind because of rape threats...Strict action against must be taken against those who threatened Kaur with rape," Atishi Marlena, a member of the AAP's political affairs committee (PAC), said.
CPI(M) General Secretary Sitaram Yechury attributed the alleged threat to "fascistic tendency" of the RSS and its affiliates.
He demanded that the central government should ensure protection to life and liberty of individuals.
Former attorney general Soli Sorabjee said, "This is scandalous. This can't be. You can't give such threats to women. This is disgraceful and must be put down with a heavy hand."
Reacting to Rijiju's tweet, Marlena said, "Who is polluting whose mind? The ABVP, through social media, is threatening women with rape. The MoS Home should not know what his priorities are when women are receiving such explicit threats."
She said, "The hooliganism which the ABVP and the BJP supporters resort to happens in the real world. But it is also happening in the virtual world.
