Simple answer: no. Neither widgets nor notifications uses a lot of battery.
However, to be clear, if you're referring to simply having notifications on (for any/all apps), then I'd assume that yes, there is some amount of battery being used to display them (e.g. notification appears on lock screen = probably-super-small amount of battery used).
If you're referring to the Weather and Stocks widgets, then no, I imagine that having them in your Notification Center uses no additional battery.