I know that the Soviets maintained a Black Sea Fleet, but I imagine that, being neutral, Turkey was not going to allow the Soviet Navy to exit, or the Kriegsmarine or Regia Marina to enter. Romania and Bulgaria I would assume maintained some sort of navy presence, but was it enough to challenge the Soviets? And did Germany attempt to built any sort of naval force to launch from its eastern allies? If so, what were they up to?
Soviet Navy dwarfed every other country at Black Sea.
However it was mismanaged, untrained and suffered heavy losses without almost single naval battle (except a few U-boat skirmishes).
Romania (the second biggest Black Sea Navy in 1941) were concentrating on protecting their ports.
Germany did build some small boats on captured Soviet shipyards and moved some small U-boats and torpedo boats by land (and Italy moved their midget submarines there as well) but most Soviet losses were from air attacks.
That said, when Stalin effectively grounded Black Sea Navy by restricting autonomous operations, there was no point for Axis to create any serious fleet there.