The answer, as it is so many times when dealing with the Seleukids, is both "we don't really know" and "it depends".

Koiné greek was obviously extended across the kingdom, usually by way of colonists and the civil administration, which we know to have used both greek and aramaic as official languages. However, the extent of day-to-day usage of Greek would've depended greatly on the place and time.

For instance, Syria, which, as you probably know, was densely urbanized and populated by greek and macedonian colonists, remained a stronghold of hellenic culture even after the fall of the dynasty and across roman and byzantine times. The same could be said about the mosaic of colonies created by Antiochos I and his inmediate successors in inner Asia Minor (Laodikeia ad Lycum, Seleukeia Sidera, Hierapolis in Phrygia); these cities, although far smaller than the ones in Syria, also mantained a relatively important status, at least on a local stance, far beyond the collapse of the Kingdom.

There were some other places across the empire with simillar characteristics: Seleukeia and its surroundings were thought to be a stronghold of 'greekness' by the Parthians -it was actually one of the reasons for the building of Ctesiphon across the Tigris-, and many other eastern cities had sizeable greco-macedonian populations: Susa, the ancient elamite capital, was, at least for a time, renamed to Seleukeia and kept an important regional position, and so did Ecbatana, modern Hamadan, in western Iran, and Hekatompylos, Merv or Herat, further east. These cities served as 'province capitals', centres of regional administration and, as such, certainly held sizeable greek populations.

Even places which had fallen off direct Seleukid control kept important greek populations: Grainger notes in his book on Antiochos III that the indian king Ashoka, grandson of Chandragupta Maurya, erected some of his famous stone edicts, namely those found in Ai Khanoum and Kandahar, in greek and aramaic -in the case of Kandahar-, which obviously means that there were enough greco-macedonian speakers around to change the languages used on purpouse. Hellenism had, actually, an interesting hold in India, thanks mostly to the campaigns of Greco-Baktrian and later Indo-Greek monarchs.

Now- it is very, very important to clarify something: urban life and rural life were tremendously different. And, despite the elevated -for the standards of the time- degree of urbanization in certain regions of the empire, the societies living within the Seleukid kingdom were almost completely rural. And there we have even less information on the degree of 'hellenization': surely, rural areas near large urban centres, such as Syria, Asia Minor or Babylonia must've seen some type of cultural shifting towards the hellenic ruling elite, but the impact of Hellenism in more remote or peripheral regions, such as the eastern edges of the Iranian Plateau or the populations of the mountains, probably was not that great, given that neither an extensive colonization proyect or an attemp of central or more direct control over said areas ever took place under the Seleukid dynasty.

All in all, Hellenism rooted mostly in and around cities and military colonies, such as Chyrros in northern Syria and Dura Europos down the Euphrates (and most of these had already become solid urban centres by the 2nd Century BCE), where Greek and Macedonian colonist populations were concentrated, while its impact was much less visible in isolated or peripheral regions, where royal grip was weaker, and the countryside.