You can change the background color of a the Event Search plugin that's embedded in a web page. You may do this either by adding the css to your web page's external css file or by adding the css directly into <head> tag of the web page containing the search widget.
Here is the css to make the background color transparent but you can replace 'none' with any hex color code.
To learn more about adding a search widget to your webpage:
Go to Admin Menu > Website Plugin > Event Search