Gutenberg certainly had a great impact on the availability of books, but his printing press didn't help bring books into the hands of the poor. The price of books dropped from thousands of days worth of wages to hundreds, but this was still far too expensive for commoners.
By the 17th century many commoners in Europe could read, but they probably had nothing more to read than the Bible which I would guess was sponsored or at least heavily subsidized to make it affordable. The poor still couldn't afford to just go to a bookstore and buy any book they fancied.
Compare it with the 20th century and onward, when books can be bought with pocket change.
When did books first become affordable to the poor? This means that even low income families could afford, if they wanted, to own several books without significant financial sacrifices. In the early to mid 19th century there were already popular authors in Europe who wrote for the tastes of the poorer classes, so I would guess it was around that time, but can this guess be improved upon?