His widely reported comments come at a time when many member states are looking to strengthen borders following terrorist attacks and a surge in refugee and immigrant numbers.
In a speech at the European Forum Alpach in Vienna, Juncker said solidarity must be shown to refugees and their children.
"Borders are the worst invention ever made by politicians," he added.
Juncker hit out at Austrians who have criticised the EU and its Schengen problems, labelling them "Landesobermuftis" or provincial high muftis.
He also defended the Commission against frequent attacks on the issue.
His remarks appear to place him at odds with Commission First Vice-President Frans Timmermans who, in a speech in January in The Hague, said, "The Schengen agreement allows border controls as emergency measures."
Speaking on the resumption of border controls by a number of member states, Timmermans stressed the importance of setting up pan-European borders and border guards, and the difficulty of patrolling sea borders compared to land borders.
Reacting to Juncker's comments, Ukip leader on the London Assembly and the party's culture spokesperson Peter Whittle told this website, "Jean-Claude Juncker's comments are beyond parody. Safe and secure borders help to define a nation, you only have to look at Germany to see what happens when you when you effectively discard them.
"This is another reason why we must exit the EU as quickly as possible, otherwise our security could be left exposed by Juncker's anti-borders policy.
"Juncker is also well behind the curve to think he, or his colleagues among the European political elite, can stop popular democracy from flourishing across Europe following the historic Brexit vote.
"I'm happy to predict that Britain will not be the only country to make the decision to leave the EU and become a free and sovereign nation again."
A spokesperson for UK Prime Minister Theresa May said, "This is not something the Prime Minister would agree with. The British people think that borders are important, having more control over our borders is important, and that is an issue we need to address."
Further comment came from Amanda Paul, of the European Policy Centre, who told the Parliament Magazine that she agreed with Juncker, adding that "more solidarity needs to be given to refugees and their children.
"So far the EU response has been pretty miserable and embarrassing for a Union that is always talking about values. I guess most member states would have been happy if Turkey were able to keep all the refugees."
Former UK Europe Minister Denis MacShane said, "Juncker might remember the old saying 'good fences make good neighbours'.
"No one wants a return to customs checks and armed police guards on every road between Luxembourg and its neighbours, but Europe's external borders are vital to the confidence Europeans today want to have.
"Ironically, the UK has never had borders with the Republic of Ireland and even when the IRA were killing and bombing in England 40 years ago, Britain never imposed border controls or demanded visas or even passports from Irish citizens. Now with Brexit all may change."
Andrew Duff, a Visiting Fellow at European Policy Centre and a former Alde group MEP, described Juncker's comments as "a charming notion."
Juncker's intervention comes after the leaders of Germany, France and Italy - Europe's three biggest countries - met on Monday to discuss the way forward for the EU in the wake of Britain's historic decision to leave.
German chancellor Angela Merkel, the French president François Hollande and Italy's Prime Minister Matteo Renzi met on the Italian island of Ventotene for talks.
Germany has reinforced controls at its border with Switzerland following a surge in the number of migrants crossing through Switzerland and a tightening in controls at the Swiss-Italy border.
The number of Syrian refugees in Europe continues to rise, but remains low compared to numbers in neighbouring countries like Turkey. Over one million Syrians made asylum applications in Europe in the five years to June 2016, according to data from the United Nations.
Following several Islamist terror attacks in France, French President François Hollande has stressed the need for EU nations to tighten borders and share intelligence.
"To have security we need frontiers that are controlled, so that is why we are working to reinforce coastguards and border guards," he said on Monday following the meeting in Italy.
Conservative home affairs spokesperson Timothy Kirkhope said: "This is clearly Junker's own view and does not reflect the opinion of most people across Europe.
"It is a rather odd comment given that a key plank of the EU's response to the migration crisis involves strengthening its external borders.
"The priority for the EU should be to distinguish the way it deals with legal migration, economic migrants and refugees. The best way to reach an agreed approach to this critical issue is not to dispense with borders but by respecting the sovereignty of individual Member States and the rights of national governments."