The helicopter crashed on Saturday on a thickly wooded mountainside around 20 km (12 miles) west of Bukavu airport.
Russia's Foreign Ministry said it was an Mi-8 helicopter owned by UTair and blamed the crash on adverse weather conditions.
A U.N. rescue team was only able to reach the crash site on Tuesday.
In a statement Ban, offered "his sincerest condolences and sympathy to the families and friends of the victims, and to the Government of the Russian Federation" and said the U.N. mission in Congo was investigating the cause of the crash.
The United Nations has an estimated 17,000 peacekeepers in Congo, and operates an extensive aviation network to counter an almost total lack of roads in the vast central African country.
The peacekeepers have been stretched thin by a rebellion in the resource-rich east of Congo and the U.N. Security Council is considering creating a special intervention force, which one senior council diplomat has said would be able to "search and destroy" the M23 rebels and other armed groups in the country.
Air accidents are common in Congo and in 2011 a U.N. passenger plane crashed in the capital, Kinshasa, killing all but one of the 33 people on board.
At least five people were killed when a twin-propeller plane crashed last week as it tried to land in bad weather in the town of Goma, north of Bukavu.