CÚCUTA, Colombia—A high-stakes effort to deliver aid into Venezuela despite a military blockade descended into violence Saturday, with pro-government gunmen opening fire on one side of the border and the National Guard lobbing tear gas at some 5,000 demonstrators attempting to rush across from Colombia.

In the end, the humanitarian aid that had been wielded by the opposition in an attempt to weaken Venezuelan President Nicolás Maduro’s government didn’t get into the country. But more than 60 National Guard forces opted to...