Simply add the following line after the view has been instantiated and set to .inputView. (It might be added elsewhere, but has to be run before the picker is shown)
UIPickerView.appearanceWhenContainedInInstancesOfClasses([UIView.self]).backgroundColor = UIColor.clearColor()
This line combined with
textInput.keyboardAppearance = .Dark and returing attributed titles from UIPickerViewDelegate's implementation that have white font will give you a dark UIPickerView which is nice if you have other UITextFields with dark keyboard.