Assuming you are using a standard Dojo theme, I would suggest that to change the colour of a tooltip background you could add the following definition to your custom CSS file.
.dijitTooltipContainer {
background-color: pink !important;
}
on quick inspection this does a reasonably tidy job of roughly re-colouring the core of the background, (the !important tag may or may not only be necessary in the quick test harness I threw up to work this out)
The pointy bit of the tooltip is defined by an image sprite though, found, in Claro, at themes/claro/images/tooltip.png (instant example from google api
http://ajax.googleapis.com/ajax/libs/dojo/1.6/dijit/themes/claro/images/tooltip.png) so one would suppose you will have to redraw that PNG in the corrosponding colour and then over-ride the background-image value of the dijitTooltipConnector CSS class.