As many people here hold the view that veil in Hindu society was introduced because of islamic onslaught, it's a necessity to write how even in pre-islamic era veil was very much prevalent in upper class of Hindu society. So, we'll be looking at some pre-islamic Hindu texts.
The earliest reference to veil is to be found in our epics. In Valmiki Ramayana, at the time when Sita Ma set out with Shri Rama for the forest through the public thoroughfares of Ayodhya+
A regret is expressed that a lady who wasn't even seen by sky spirits should now become the object of public gaze 
A similar incident takes place in the Mahabharata at the time of Dhritrashtra's departure to forest. It's stated that the women who hadn't seen either sun or moon, came out into the streets on that occasion, in great grief, when King Dhritarashtra proceeded towards forest.
In Valmiki Ramayana, there's a detailed description of mourning of Mandodari along with other wives of Ravana. During Mourning, Mandodari asks why her husband isn't enraged seeing her and his other wives unveiled.
In Bhasa's PratimāNātaka (c. 200 AD), we see Sita Ma coming on stage with a veil, but Shri Rama asks her to remove it in order to allow the weeping citizens of Ayodhya to have a parting glance. Even Dasaratha's widowed wives are shown going out of palace with a veil.
In The Mrichchhakatika of Sudraka, we find Vasantsenā being offered a veil, when she was raised to the status of a high-class woman after marrying a Brahmana.
In Kalidasa's Shakuntala, when Shakuntala was taken to King Dushyant, he describes her being in a veil. When Dushyant doesn't recognise her, Gautami even comforts Shakuntala by telling her that Kind Dushyant will definitely recognise her after she removes her veil.
In Lalitavistara, when Buddha's bride-elect Gopā is shamed for not wearing a veil. She even criticizes those shaming her. So, we can atleast assume that the practice of veil was prevalent enough to be criticized.
To conclude, high class women of Hindu society led a sheltered life. This is shown by the term "Antahpuram", i.e, inner apartment. Strangers were not expected to enter it. Women themselves would often retire into seclusion, when they found out their natural guardians weren't with
Them. In Svapnavāsavadattā of Bhāsa, queen Padmavati observes that it was but natural for Tapasi to avoid Strangers, because her husband was out on a journey.
Britishers in UP and Bihar noticed that only upper class women used to cover their faces. While it was never observed among the lower class.
Smritis like those of Manu and Yājnavalkya lay down detailed rules about the life of women, but they nowhere mentioned any strict rule for women to cover their faces. So, the practice must have been a thing of choice rather than an orthodox rule.
Nothing related to Hindu society but to get some perspective we can look at some pre-islamoc civilizations too. In Athens (c. 500 BC) women could not meet their husbands' guests or go outside without proper guards. A Bridegroom couldn't see his bride before marriage.
In Sparta, women had separate apartments and could not see be present at a banquet.
In Ancient Assyria, veil was worn by married women while harlots and concubines were regarded as ineligible for it.
