Games were not normally free. Sometimes emperors would sponsor events as a way of winning popularity with the public. Julius Caesar was, I believe, the first to do this.
Late in the empire the emperor subsidized games out of the public purse.
Women did attend. The Romans had no problem with women appearing in public. In fact, not only did women attend, sometimes well known free women actually fought in the arena for money or fame, even though it was considered "disreputable" to do so.
Fine. Force me to be the professor. Reading materials for the student:
The Story of Civilization volumes 4 and 5 have articles on Roman gladiatorial games.
Encyclopedia Britannica volume 11, article on gladiators
Among modern works is "Gladiators and Caesars: The Power of Spectacle in Ancient Rome"
by Eckart Köhne, Cornelia Ewigleben, Ralph Jackson. In this book, it proves my point that entrance money was charged in many cases on page 20 where it says:
[Private organizers] derived high profits from the entrance money...
As it says in the answer above, even when the games were "free" because some famous person like Caesar was paying for them, there were still tickets required and those tickets were resold by scalpers, thus making the "free" event actually quite costly to attend.