I'm talking about the era during which the translations were made. The factions behind Geneva and KJV would go on to wage the English Civil War, while Douay-Rheims was much reviled and many Catholics were arrested or driven out of the country. It was just a volatile period, and each faction had its "correct" version and condemned the other versions. At the same time, Geneva and KJV borrow liberally from common sources (previous translations) and KJV borrows from Geneva and from Douay Rheims -- the King's translators would never have admitted it, but a comparison of the texts shows they did envy the Jesuit scholarship, especially in Hebrew. Maybe I'm not telling you anything you didn't know.
As for nowadays, there are a couple of Reformed denominations that have adopted the Geneva as the "correct" version. I think the Federal Covenant (or however the name goes) is one. It's probably a reaction to the IFB 1611ers. Churches be crazy out there.