More than 10 incidents occur daily, local group says, in a country reeling from economic crisis and high inflation.
Tens of thousands of Venezuelans have poured across into neighbouring Colombia to buy basic goods amid shortages at home, during a brief opening of the border that has been closed for almost a year.
Venezuelan President Nicolas Maduro closed the border in August 2015, citing smuggling and penetration by Colombian fighters. On Sunday, he authorised a 12-hour opening of the pedestrian bridge that connects Tachira, in Venezuela, and Cucuta, in Colombia, for the first time in 11 months.
Victor Bautista, the director of the Colombian border authority, said that an estimated 35,000 Venezuelans had arrived in Cucuta.
Some 25,000 people surged in within the first seven hours, William Villamizar, governor of the Norte de Santander department that includes Cucuta, said on Twitter.
“Thank you for this welcome, the people of Venezuela are experiencing a serious humanitarian situation,” Jose Gregorio Sanchez, a resident of the border town of Urena, told the AFP news agency.
Venezuela has been mired in a deepening economic crisis that has emptied shop shelves and created a shortage of medicine. The recent slump in oil prices devastated the OPEC nation’s economic model, leading to growing anger among the roughly 30 million residents.
Critics also blame grave mishandling of the state-led economy.
Maduro, elected in 2013 after the death of President Hugo Chavez, insists that he is the victim of an “economic war” led by businesses with the backing of the United States.
Protesters demanding food have clashed with authorities in several cities in recent weeks amid demonstrations and looting that have turned deadly.
Some 500 desperate Venezuelans illegally stormed the border earlier this week in search of basic goods.
“There’s no medicine for children; children are dying,” Tulia Somaza told AFP.
“People don’t even have soap to wash clothes.”
In a sign of Maduro’s concern at mounting social unrest, the president replaced the head of the National Guard on Thursday.
The Venezuelan opposition launched its efforts to remove the president, including a bid for a recall referendum, after winning control of the legislature in January.
But Maduro has challenged his rivals through the Supreme Court, which they accuse him of controlling.
Maduro ordered the border shut last year after former Colombian fighters attacked a Venezuelan military patrol and wounded three soldiers, causing a diplomatic row between the neighbouring countries.