I am going to limit this response to the 5th and 4th centuries BCE, the heyday of Athenian democracy. In this time, dramatic performances were part of two main festivals to Dionysus, the "City" Dionysia (around the vernal equinox) and the Lenaia (in the month of Gamelion, roughly January). That is--there were not performances every day. These were special occasions and were a part of larger, week-long holidays.

Early on, the City Dionysia was more for tragedy (and other dramatic performances like satyr plays and dithyrambs) and the Lenaia was more for comedy, but this distinction becomes more muddled towards the end of the 5th century. There was also a "Rural" Dionysia, at another time of the year, which in the larger demes had some dramatic performance (often re-run of previous year), but let's stick to the theater activity in the astu (the city of Athens herself).

The theater experience was thoroughly democratic, and was open to all (primarily male) citizens. The poorest citizens in the most extreme corners of Attica perhaps did not make it to the city every single year, but the impression we get is of a well-attended event by a relatively representative quorum of the citizenry. The venue was the Theater of Dionysus, right under the Acropolis, which by the middle of the 4th century could hold roughly 17,000 spectators, and most of those seats were reserved for Athenian citizens. In fact, the theater was one of the major venues of democracy, along side the ekklesia (assembly of the people), the boule (the small council), and the dikasteria (courts). Although the Athenians were scared away from tragedy on contemporary events early on, comedy ("Old Comedy" of the 5th century) was extremely political in nature. Aristophanes wrote plays about the war, about the annoying habits of his fellow citizens, and about his extreme dislike for certain Athenian politicians. His Knights is an incredible production aimed specifically at a demagogue named Kleon. The whole play is in fact very apt for the modern world.