For the Roman empire, there really was no "border", at least not in the sense of a modern nation-state. Theoretically, Rome's imperium had no terretrial limit: the barbarians were, on paper, under Rome's authority as well: they simply weren't incorporated into the provincial organization of the empire. Of course, in reality that wasn't the case. Anyone actually living in the frontiers would know that if you kept walking away from Rome, at certain point you were in barbaricum outside of Rome's control. Especially in the East, no one would have been delusional enough to believe that Parthia/Sassanid Persia was a Roman subject. But when would that person know that? For the frontiers along the Rhine, the Danube and the Euhprates, the great rivers were both religiously symbolic and visible signs. All Roman forts were on one side of the river, so crossing would have been a good indicator of stepping into the "other side" as well. Still, trading, traffic and military patrols between both sides were very common, so it's difficult to say just how much of a big deal that was, although Roman writers like Tacitus did see the great rivers surrounding and bounding the empire (Tacitus, Annals 1.9). Things would have been even more blurry in the North African and Arabian frontiers, which didn't have any rivers as a visible sign of demarcation. It's probably more easier and accurate to perceive the "border" between the Roman empire and the barbarians as a zone rather than a linear line.

That does not mean however, that the Romans had no sense of linear demarcation. Indeed, the very idea of a boundary was an integral part of the foundation myth of Rome, symbolized by the boundary (pomerium) of Rome ploughed by Romulus.

There are many examples of termini stones, or border markers, between individual provinces and cities. Such markers would be an easily recognizable sign of demarcation.

It is also important to note that there were many different layers of borders within the empire. The administrative, judicial, economic and military borders were all placed separately inside the empire. For example, the portorium, the dozen or so tariff/road toll zones of the empire, were set up independently from the provincial boundaries. In some areas, it would have been possible to walk into a different tariff zone even if you were within the same province.

TLDR: there really was no linear "border" in a modern sense in the Roman empire. The border was a lot more zonal and gradual. The great rivers along some of the frontiers was both a symbolic and visible sign of demarcation, but for frontier areas without the border would have been even more blurry. However, there were clear demarcations within the empire, which were administratively, judicially, economically and militarily set up independently.

Sources

: David Braunt, "River frontiers in the environmental psychology of the Roman world", in David Kennedy (ed.), The Roman Army in the East. Journal of Roman Archaeology Supplementary Series no. 18, (Ann Arbor, 1996), pp. 43-7.

: Hugh Elton, Frontiers of the Roman Empire (Indiana UP,1996)

: Boris Rankov, "Do Rivers Make Good Frontiers?" in Zsolt Visy (ed.), Limes Xix. Proceedings of the Xixth International Congress of Roman Frontier Studies Held in Pecs, Hungary (September, 2003), (University of Pécs, 2005), pp. 175-81.

: C.R.Whittaker, Frontiers of the Roman Empire: A Social and Economic Study (The John Hopkins UP, 1994)

: C.R. Whittaker, Rome and its Frontiers: The Dynamics of Empire (Routledge, 2004)