It all depends on what countries you are talking about. There are more slaves now than in any time in history.

In Britain's case, the late eighteenth century saw the rise of the Romantic and the Evangelical movements. The Romantics placed feeling above calculation and the Evangelicals burned with intense zeal for righteousness and had a highly self-conscious regard for morality. Thus the slave trade struck the abolitionist movement (largely Christian moralists) as an abomination. The abolition of slavery was the first great victory for this movement that would come, during the Victorian era, to be heavily influential. They began as a minority and by the mid-nineteenth century were a majority.

I know! I'll be called down on this. The question just begs for a point of view approach to an answer. When did people start to think slavery wasn't cool? From the Slaves point of view...probably immediately!