As in, read and appreciate them apart from the actual theatre? Were they printed/sold? If not, when did reading plays as works of literature became common?
Shakespeare's plays were bound and printed individually -- and eventually collected in 1623 in the First Folio -- and it's obvious that many of the extant Greek tragedies and Roman plays were also printed and published (to say nothing of the numerous Chinese, Japanese, and, yes, even "American" plays being written and distributed around this time.)
It's most likely that plays took on a literary value around the time of the Industrial Revolution, when theatre attendance and participation itself began to be democratized by the rise of the leisure class -- the middle class, with money, who could suddenly afford to attend social events at their leisure.
About half of people in England at the time were illiterate, so they only went to see plays. Shakespeare's plays were printed without his permission and distributed, and his poetry was quite popular, so his plays were probably read pretty widely as well, at least amongst those who could read.
In general people didn't really read plays, they saw them as "ephemeral" entertainment that couldn't really be captured through print.