There is no general brilliance control provided in Picasa. You can increase highlights and/or add fill-in light to increase brilliance, You can also make the dark areas darker by adjusting the shadow slider, but this does not effect the bright and mid tones.
However you can achieve overall darkening by using the graduated filter. This is generally used to decrease the brightness of the sky and enhance blue colour or cloud definition. In this effect you set where the graduation is to end which is indicated by a cross which is set at the centre of the photo, the amount of shade and feathering.
If you drag down the cross to the bottom of the photo and set the feather to 0, then the whole image is darkened uniformly and the amount can be controlled by the shade slider.
You can also change the colour of the shade, but being colour blind I keep away from this feature.