Political campaign designer Zac Damon, who worked for Vote Leave and UKIP, left Swinson and her husband so terrified over their son's safety they had to put special measures in place at his nursery.
The 40-year-old yob, who lives in Portsmouth, sent menacing emails to the mother of two, threatening: 'you might want to take extra care of that little son of yours’ in one.
A judge said the shocking threats reflected how some used the Brexit campaign 'in the most obscene way' to attack people with different opinions.
Swinson, who received the emails during her four-and-a-half month tenure as Lib Dem leader, said she 'felt sick' when she opened the messages at Parliament and 'all I wanted to do was hug my child close'.
The 40-year-old revealed she was 'used to abuse' but 'a threat to children is so past the line' and said her husband Duncan Hames was 'looking over his shoulder' as he walked their then one year old son to nursery.
A judge condemned Damon and jailed him for 12 weeks. District Judge Tan Ikram said: ‘This was a threat to a high profile figure whose position on Brexit was clear and you were working on the other side.’
Damon designs logos and social media posters for political campaigns, stating Portsmouth North Tory MP Penny Mordaunt as one of his clients.
Portsmouth Magistrates' Court, heard that on September 9 last year Damon sent Swinson a foul-mouthed email in which he branded her ‘utterly pathetic’.
On September 25, he sent another. It read: ‘You might want to take extra care of that little son of yours, if not now when he begins school in the future.’
Prosecutor Rhys Evans said: ‘Damon said he sent the second email as he was genuinely worried about threats he saw online, but by virtue of him changing his not-guilty plea to guilty, he has abandoned that line of defence.
‘It clearly caused her and her family substantial distress and fear.’
Swinson, in a victim impact statement, said the murder of MP Jo Cox 'changed' how she believed she should deal with the abuse she receives.
Swinson said she even had to visit her child's nursery to tell them about the threat and have security do a sweep of the site.
She added: ‘Receiving this was chilling, I felt sick. I was at work in parliament at the time and all I wanted to do was hug my child close.
‘I didn't know if they were deliberately planning to target my child.
‘My husband describes the horrible feeling of looking over his shoulder when he was walking our child to nursery.
‘I thought I was pretty used to the abuse I have received over the years but this was difficult.
‘It strikes at the very heart of wanting to protect your children. A threat to children is so past the line.’
Judge Ikram, also giving Damon a two-year restraining order preventing him from contacting Swinson, said: ‘It can't be said that you are a man who has shown regret.
‘You were working for the Conservative, Brexit and UKIP parties, for their social media and posters. You knew the power of messaging because that's the industry that you are in.
‘It does not matter what point you were trying to make, it's the profound impact that it had on the victims.’
The judge said politicians ‘sadly’ accepted they could be called names and abused but insisted the second email had been a threat.
‘This must be seen in the context of a politician seeing a colleague murdered,’ Judge Ikram told Damon.
‘It is in the context of the Brexit campaign being used and abused in the most obscene way to abuse people who don't agree with what you say.
‘Politicians are public servants who devote their life trying to make the world a better place. Threats like this deter good people from entering public service.’
Simon Moger, defending, said Damon 'wants to extend his apologies to Swinson and her family'.
Mr Moger said Damon was 'upset that he caused distress and is more mindful now' but the judge refused to accept his suggestion this was 'impulsive' behaviour.
Damon will serve less than half his 12-week sentence.