Fears also mounted for elected President Mohamed Bazoum, who was ousted by members of his guard on July 26, with reports saying his detention conditions were deteriorating.
Protesters near the base on the outskirts of the capital Niamey shouted "down with France, down with ECOWAS," a reference to the West African bloc which on Thursday approved deployment of a "standby force to restore constitutional order."
Many brandished Russian and Niger flags and yelled their support for the country's new strongman, Gen. Abdourahamane Tchiani.
"We are going to make the French leave! ECOWAS isn't independent, it's being manipulated by France," said one demonstrator, Aziz Rabeh Ali, a member of a students' union.
Former colonial power France has around 1,500 personnel in Niger as part of a force battling an eight-year-old jihadist insurgency.
It is facing growing hostility across the Sahel, withdrawing its anti-jihadist forces from neighboring Mali and Burkina Faso last year after falling out with military governments that ousted elected leaders.
Niger's new leaders scrapped defense agreements with France last week, while a hostile protest outside the French embassy in Niamey on July 30 prompted Paris to evacuate its citizens.
Fears for Bazoum
The European Union and African Union joined others in sounding the alarm for Bazoum on Friday.
"Bazoum and his family, according to the latest information, have been deprived of food, electricity and medical care for several days," said EU foreign policy chief Josep Borrell.
United Nations rights chief Volker Turk said Bazoum's reported detention conditions "could amount to inhuman and degrading treatment, in violation of international human rights law."
The AU echoed the concern, saying "such treatment of a democratically elected president" was "unacceptable."
German Foreign Minister Annalena Baerbock warned that the "coup plotters must face harsh consequences should anything happen" to Bazoum or his family.
A source close to Bazoum said "he's OK, but the conditions are very difficult," adding that the coup leaders had brandished the threat of assaulting him in the event of military intervention.
Human Rights Watch said it had spoken to Bazoum earlier this week. The 63-year-old described the treatment of him, his wife and their 20-year-old son as "inhuman and cruel," HRW said.
"I'm not allowed to receive my family members (or) my friends who have been bringing food and other supplies to us," the group quoted him as saying.
"My son is sick, has a serious heart condition, and needs to see a doctor," he was quoted as saying. "They've refused to let him get medical treatment."