From my limited reading on the subject, it seems that the Chinese were aware that something happened to Da-Qin (meaning "Great China" = the old Roman Empire). According to them, the most notable sign was that (starting in Late Antiquity) regular envoys stopped being received in the Chinese court, casting some doubt as to the survival of the Da-Qin. Occasionally, one of the Emperors would send an envoy (as far as I know only documented in the Chinese records -- yet, the vast volume of obscure and unstudied Byzantine documents might still yield something), but it would often be decades or centuries between a visit (especially following the Arab Invasions). Whether they knew that the Empire "fell" is up to debate, but it seems, indirectly, they were aware that something drastic had occurred.

However, one large clue as to their understanding of the situation can be observed thusly: starting in the 7th Century, Chinese records curiously start using a new term, Fu-lin (apparently meaning "wealthy successor", see below for further discussion) to describe what is to be seen as the same Empire (Da-Qin) that they had communicated with in the past. Indeed, a number of medieval Chinese writers draw a direct connection between Da-Qin and this new Fu-lin. This could reveal that the Chinese saw enough of a change in the nature of the Da-Qin to warrant creating a new name for them, or that the relation to Da-Qin was questioned. However, from the descriptions they provide of this spontaneous new country (which are quite detailed), it seems to me (and others) that Fu-lin is the Byzantine Empire.

Chiu-t'ang-shu [The Old Tang History], ch. 198 (c. 10th Century):

The country of Fu-lin [Byzantium], also called Ta-ts'in [Roman Syria] borders on Po-si [Persia]. Its territory amounts to over 10,000 li. Of cities there are four hundred....The walls of their capital are built of granite, and are of enormous height [the Theodosian triple walls]. The city [Constantinople] contains in all over 100,000 households [some 500,000 to 600,000 inhabitants]. In the south it faces the great sea. In the east of the city there is a large gate; its height is over twenty chang [over 235 feet]; it is beset with yellow gold [bronze] from top to bottom, and shines at a distance of several li. Coming from outside to the royal residence there are three large gates beset with all kinds of rare and precious stones...

Hsin-t'ang-shu [The New Tang History], ch. 221 (late 11th Century)

Fu-lin [Byzantium] is the ancient Ta-ts'in [Roman Syria]. It lies above the western sea [Indian Ocean]. Some call it Hai-hsi-kuo [i.e., "country on the west of the sea"]. It is 40,000 li distant from our capital and lies in the west of Shan [Armenia]; north you go straight to the Ko-sa tribe [Khazars] of Tu-ch'ueh. In the west it borders on the sea-coast with the city of Ali-san [Alexandria]. In the south-east it borders on Po-si [Persia]. Its territory amounts to 10,000 li...During the 17th year of Cheng-kuan [643 C.E.] the king Po-to-li [Constans II Pogonatus, Emperor 641-668 C.E.] sent an embassy offering red glass and lu-chin-ching [green gold gems], and a cabinet order was issued as an acknowledgment. When the Ta-shih [Arabs] usurped power over these countries, they sent their general, Mo-i [Mo'awiya, then Governor of Syria, afterwards Caliph 661-680 C.E.], to reduce them to order. Fu-lin obtained peace by an agreement, but in the sequel became subject to Ta-shih.

From the Ming-shih, ch. 326 (concluded 1724 C.E.):

Fu-lin [Byzantium] is the same as Ta-ts'in [Roman Syria] of the Han period. It first communicated with Zhongguo [China] at the time of the emperor Huan-ti [147-168 C.E.]. During the Chin and Wei dynasties it was also called Ta-ts'in, and tribute was sent to Zhongguo. During the T'ang dynasty it was called Fu-lin. During the Sung it was still so called, and they sent also tribute several times; yet the Sung-shih says that during former dynasties they have sent no tribute to our court, which throws doubt on its identity with Ta-ts'in.

Translations taken from Fordham University's Internet History Sourcebook. You can read the entire source excerpts regarding relations between Rome/Byzantium and China on this page -- highly recommended.

EDIT: Please read below for discussions on nuances that may need to be further considered.

I think this really depends on what you consider the fall of the Roman Empire. The Roman Empire exists as a continuous state until 1453 when Constantinople fell. This would be during the Chinese Ming Dynasty and they may have noticed a drop in trade, there were many issues between the Europeans and the Turks (which lead to Spain and Portugal looking for other trade routes to the east). That said, the Ming Dynasty is most certainly not ancient China.

If you are looking for if the Chinese noticed the fall of the Western Empire, I would say probably not. There was still the Eastern Empire which was larger and more prosperous, and the fact that the Western Empire fell very slowly.

So I'm going to jump into this debate sideways, but still tagging /u/shlin28 and /u/ambarenya and cite Pulleyblank's article "The Roman Empire as Known to Han China", where he points out that the main problem of China's knowledge of the Roman Empire is we can't exactly separate their idea of Da Qin as a territorial "Roman Empire" from Da Qin as a fantastical "Nega-China", a distant land that is viewed as a polar mirror to their own. It was a region so far away that the normal ideology of civilization radiating outward from ZhongGuo/Central Kingdom/"China" ceased being applied.

This is what really should be understood when you conceive of "vague awareness" between the two states, going from the direction of China to the west. It is vague to the point of not even being discernable from fable.

Once you get past Han China, into Tang China, with their renewed control of the Taklamakan basin, then Fu-lin crystalizes into actual focus for the Chinese as a physical region to which ambassadors could be sent. However, I would treat matchups of Chinese names to their known western counterparts, like Fu-lin with Byzantium as suspect, and possibly refutable at some point in the future.

As quite powerfully demonstrated in that Pulleyblank article, there is a multiplicity of layers, each with the possibility of misidentification, in associating a Chinese region with its western counterpart. At the base level, you have to deal with whether or not the associated names were phonetically transcribed or some kind of idea representation. Above that, if phonetic, you have to question whether you're dealing with Old Chinese (which is highly contested in its reconstructions) or Middle Chinese (which sounds very little like Mandarin). If an idea representation, it gets worse because the Chinese could've very easily associated particular place name as a generic "eastern city." Pulleyblank gives a breakdown of a chinese name frequently associated with an eastern Mediterranean city (I think it was Antioch) as actually a region in northern Iran because of sound-shifts. And even if Fu-lin was ultimately associated with Byzantium later on, once the lines of communication solidified around the Mongol era, you have no guarantee that it is the same Fu-lin the Chinese were thinking about in their 7th century embassies. The Imperial court was not exactly in need of such exacting identification of states so liminal to them.

So basically, I would take all information on Sino-Western communication as suspect, until around the 10th-11th century.

As for whether the Chinese knew of the Roman Empire's fall, because of Da Qin's near-mythological status in Chinese mind, I would say they did not know if its "fall" because the Byzantine state was around, but with one big caveat.

Da Qin was the name used for the Nestorian Church in Tang Chang'an in the 8th century. The better question to ask, is whether there was a decline or shift in the usage or understanding of "Da Qin" in all its mythic guise, in China than necessarily applying a 1:1 label with its decline to the fall of WESTERN Rome.