Note that style modifications apply to all the tooltips on the page.
Dojo creates (when required) 1 common dijit._MasterTooltip object that holds tooltip's dom nodes, and all tooltips use it.
The following code adds custom css classes to the rooltips's container and connector.
if(!dijit._masterTT)
dijit._masterTT = new dijit._MasterTooltip();
// Add a css class to the container
if(dijit._masterTT.domNode.childNodes[0])
dojo.addClass(dijit._masterTT.domNode.childNodes[0], "classContainer");
// Add a css class to the connector
if(dijit._masterTT.domNode.childNodes[1])
dojo.addClass(dijit._masterTT.domNode.childNodes[1], "classConnector");