If you define the Romans as citizens of the Roman Empire who most likely considered themselves to be Roman, the answer is: certainly yes. The southern part of the Arabian Peninsula and the Horn of Africa have been integrated into the economic and cultural network of the Mediterranean during the course of the first millenium BC and later. Thus, the material remains of this area known as the Kingdom of Sheba (home of the biblical figure of the Queen of Sheba) show very distinct similarities to that of the greek and roman cultural setting in the north. While this doesn't necessarily implies the actual presence of greek and roman actors, it's highly probable that at least merchants were present. Very clear evidence comes from the city of Aden (or Eudaimōn Arabia in latin): Its importance as a trade port is mentioned in the 2^nd century AD by the graeco-roman geographer Ptolemy and church historian Philostorgius tells us that a church was build in Aden during the reign of Constantius II in the middle of the 4^th century AD.

While the Horn of Africa and “Yemen” have always been connected with the Ancient Near East, and therefore to the Mediterranean world (thanks to the Red Sea), relations with (i) western Africa and (ii) areas south of Ethiopia were not as self-evident. Before the Romans, we have two accounts of Phoenician ventures in these regions (see the alleged circumnavigation under Necho II, c. 600, and the more probable expedition by Hanno of Carthage, c. 400). But the early Roman Empire also ventured in these uncharted areas, especially in the first century BC, the “golden age” of the Principate, when the relative lack of external threats made such curiosity possible. Pliny is our main source for these occasional forays. There has been considerable (and often idle) discussion over the actual breadth of the area he describes — many of the topo- and ethnonyms he uses do not find any modern correspondance, which has given way to many wild extrapolations. However (I might be more precise later on if I find the time), it is clear that the Romans reached eastern Africa south of the Horn (the coast of modern Tanzania), but also the White Nile (modern Sudan/Uganda) and a part of western Africa (modern Mauritania, Senegal). The most original element of their attempt is that part of these explorations (western Africa, Nile) were made on foot, and not exclusively through coastal navigation.