Latin was still an important language in the sixth century, but we really don't know how much Latin an 'average' citizen knew. There was definitely a large Latin-speaking population in the empire, particularly in Illyricum (hence why Justin I and Justinian were native-speakers of Latin), and even more after Justinian's wars, as it brought North Africa and Italy back into the fold. In the capital there was a large Latin-speaking community, but we only really know about the wealthy and the influential, such as exiled aristocrats from Italy and Africa, though Greek nobles and bureaucrats would be trained in Latin too. Some of the poor/middling classes spoke Latin, as we know many poor Illyrian villagers fled to Constantinople to join the army. Justin I was one such man, being successful enough to eventually become the emperor despite his illiteracy. Overall though we don't have any idea of the numbers involved apart from the guess that Latin-speakers formed a reasonably-sized minority in the city. Beyond that, all we know is that Latin writers continued to work at court and were often rewarded for their literary compositions, with John Lydus and Corippus being two great examples, as both received imperial patronage even though the former was most likely a pagan!

Even so, Latin was declining in some ways already. Though we know of many Latin writers, they were undoubtedly only a minority amongst Latin-speakers, and Latin-speakers themselves would have been dwarfed by the number of Greek-speakers overall. John Lydus for instance complained that even the educated classes were abandoning the language (though it might just be a rhetorical device to emphasise how awesome their Roman past was). Likewise, whilst the Justinianic Code was initially compiled in Latin, Justinian's laws from 541 onwards were only issued in Greek, whilst inevitably the vast majority of the bureaucrats working in the empire would be native Greek-speakers using Latin only for official purposes, which must have speeded up the process of decline. Latin did not die quickly though, as Gregory the Great in the late sixth century recorded that even though his Greek was terrible, he made many friends in Constantinople and engaged in theological debate there when he served in the city as a papal ambassador.

The seventh century however was an extended crisis for the empire and saw the loss of most of its influence in the west, so I think it would be reasonable to argue that the Latin population of Constantinople started to use Greek more and more during this chaotic period, though it might be alleviated somewhat by all the refugees running from the Avars and Slavs now dominating the Balkans. Realistically though, in an empire that was increasingly using Greek only they would have adapted to using Greek quite quickly. Bilingualism was still common amongst the educated classes though, since from the few sources we have we know that many prominent monks and imperial officials travelled across the Mediterranean to rally support for and against certain religious doctrines, with Rome and Carthage being two focal points, so a Latin education was still widely available for those who wanted/needed it.

As a follow-up: When did Greek become incomprehensible to the average person in Constantinople/Istanbul?