Becky is the head servant to the Locktons in New York. She's mainly in charge of the cooking, but she also decides which work is to be done by which slave or servant.
Explaining exactly how Becky feels about the war is a much more difficult answer. Becky understands that she works for awful people; however, Becky also knows she works for a very wealthy and influential family. She doesn't want to do anything to jeopardize her standing in the Lockton household. Because of that motivation, Becky tries not to let her feelings about the war show.
Chapter seven contains a small exchange between Becky and Isabel that somewhat shows how Becky feels about the war. Becky is explaining to Isabel how to do some of the work around the house. Three Patriot soldiers walk past, and Becky says the following:
"I wish they'd all go home. . . Soldiers is a nuisance."
Isabel thinks Becky is a Loyalist because of the comment, and asks for further clarification. Becky quietly explains that the Locktons are Loyalists. As a servant of Loyalists, Becky says she is a Loyalist in public as well.
"Listen to me good. Them that feeds us" — she pointed upstairs — "they're Loyalists, Tories. That means we're Tories, too, understand?"
Based on that comment, I think Becky secretly supports the Patriots, but she isn't willing to risk her station in order to publicly support her feelings.