|Flag of the Soviet Air Force -Wikipedia|
Looking at open-source satellite imagery it was obvious that the airport was, or had been, military from the alert aprons with hard stands surrounded by berms at the end of the runway. But did they have nuclear weapons? Did this base house some of the Soviet Long Range Aviation force? No to both. This was a fighter-interceptor base during the Cold War with MIG-31 (МиГ-31) and MIG-25 (МиГ-25) aircraft, part of the 83rd Fighter Aviation Regiment, not part of Long Range Aviation; therefore no nuclear weapons would have been in storage.
The airfield is still very much in use; I believe by the 535th Mixed Aviation Regiment and 546th Army Aviation Air Base. Photos are readily available on PlaneSpotters.net showing the variety of aircraft that operate from the airport. According to Jane's, the 535th flies the An-26, An-72, An-26, and L 410UVP while the 546th flies a variety of attack, assault and transport helicopters such as the Mi-35M, Mi-24P, Mi-28N, Mi-8, Mi-26 and Ka-52 (shown in below embedded video).
If you are not familiar withe Russian geography, here is where Rostov-on-Don is located
Victory Day Day Parade / Air Show over Rostov on Don May 9th 2015