Directly? No, not that I know of. There is some suggestion that it may have been used in a northern route of the overland Silk Road, but it isn't particularly well understood. The only real examination I have ever seen of it is in Raoul MacLaughlin's Rome and the Distant East, which necessarily relies heavily on supposition and comparative evidence.

As for what they thought of it, this is really rather tricky. The Romans only tenuously and briefly commanded a direct foothold on the coast, and it doesn't see to have excited that much interest in what authors we have. Eratosthenes in second century BCE Alexandria believed that it was a bay in the "surrounding sea" believed by the Greeks to encircle the land of the earth, and this interpretation seems to have been followed by Pliny the Elder. Ptolemy, writing in the Alexandria of the second century CE, understood it was an inland sea. The increase of knowledge is likely a reflection of greater Roman involvement in the region in the early second century CE.

Unfortunately, your interesting question is not really something I am aware we can answer in any real way.

To the best of my knowledge, the Romans never directly controlled territory fronting the coast of the Caspian, as that was largely within the territory of the various Persian empires (Parthians or Sassanians) or the loosely-organized Steppe polities (Huns, Scythians, etc). I think some roman client states like Armenia or Colchis may have controlled some area of black sea coast-line at some point, but that would have been more Late Antiquity (400-600 AD) than during the heyday of the Roman Empire in the 1st-3rd centuries AD.