During the Gilded Age how were the people denied democratic self government?
I would argue that it is a bit overstated to say that people were denied democratic self-government during this time, but it is safe to say that they had less of it than we modern people would think they should have had. This was largely because of the corruption that pervaded this era.
During this time, the "robber barons" were making their money and were using part of that money to buy off people in government. It was during this time that the Senate (which was not popularly elected in those days) was seen as a group of men who had been "bought" by the big business interests. You can see many political cartoons from those days which show this image of the Senate. At the same time, there were all sorts of corruption scandals in the government. A major example of this was the Grant Administration, which was rocked by many scandals such as the Credit Mobilier scandal.
Because there was so much corruption, you might argue that people did not have as much control over the government as they ought to have had. In that way, you can say they were denied democratic self-government.