There's going to be a great deal of variability in terms of location, even within the West, but I actually just wrote a section on money in Byzantine Southern Italy for my dissertation.

In that area (roughly equivalent to modern Calabria, Basilicata, and Apulia), your average farmer/non-urban dweller would have interacted with physical money on a regular basis at least since the 9th century. Byzantium used a trimetallic system (copper folles, silver miliaresia, and gold nomismata, as we all as gold tetartera later on. The system changes completely at the end of the 11th century), though silver coinage seems to have been strangely absent from Italy. The copper coins may have been used by these villagers and farmers for ordinary purchases, but even if they did not regularly spend folles they would have handled the gold nomismata regularly for taxation purposes.

Taxes were required to be paid in gold and were assessed based on territorial divisions, and the people living in this area (a chorion, which is also the term for a village) would be responsible for a set amount of taxes based on the value of the land as assessed by Constantinople. Some lands were also designated as military lands and would be responsible for a given number of soldiers and the money necessary for their equipment and provisioning (known as the strateia). Because these taxes were required to be paid in gold, there were presumably money changers available to swap the villagers' copper coinage for gold to be paid to the government since it's unlikely that people had gold coinage on hand given their value.

Based on the documentary evidence as well as the number of coinage finds, it seems that money was pretty ubiquitous in the Byzantine world, including their western territories, by the 9th century (this would have been the case in the late antique world as well, but there is a break in the monetization of the empire during the crises of the 7th and 8th centuries. Whether people living in the countryside would have had a supply of coinage with them at all times is impossible to say, but having and spending physical money would not have been an uncommon practice.

Hope this helps, I'm curious to hear other answers for other parts of Europe!

Sources

-Cecile Morrison, "Byzantine Money: Its Production and Circulation" in The Economic History of Byzantium.

-Lucia Travaini, La Monetazione nell'Italia Normanna (Rome, 1995).