The group includes 12 Poles and one Dutch citizen, state news agency PAP reported.
Polish Interior Minister Mariusz Kaminski called the attempt to destroy the barbed-wire barrier "absolutely unacceptable" and vowed to bring all those responsible to justice.
Meanwhile, Defense Minister Mariusz Błaszczak said that "those who destroyed the fence are joining a hybrid attack on the eastern border of Poland" and the European Union.
"Such behavior threatens the security of our homeland, especially on the eve of the Zapad 21 exercises,” Błaszczak added.
Russian-Belarusian military exercises dubbed Zapad 2021 are due to be held near the Polish border from September 10 to 16.
About 200,000 soldiers from the two countries are set to take part, public broadcaster Polish Radio’s IAR news agency reported.
In recent weeks, Poland and Lithuania have seen increasing numbers of migrants from countries including Iraq, Syria and Afghanistan attempting to cross from Belarus.
Błaszczak warned last week that the developing migrant crisis was a result of "a hybrid war, and an attack on Poland."
He announced at the time that his country would deploy up to 2,000 soldiers to protect the border.
The prime ministers of Poland, Lithuania, Latvia and Estonia said earlier this month that the migration crisis “has been planned and systemically organized by the regime of Alexander Lukashenko.”
The joint statement by Poland's Mateusz Morawiecki, Lithuania's Ingrida Šimonytė, Latvia's Arturs Krišjānis Kariņš and Estonia's Kaja Kallas was released after Poland and the Baltic states last week accused Lukashenko's regime of intentionally pushing thousands of migrants over the EU's eastern border.
Click on the audio player above to listen to a report by Radio Poland's Agnieszka Bielawska.