Look lower. The delicious widget UL gets an ID of “delicious-list” via this bit of PHP code:
ul.setAttribute('id', 'delicious-list');
If you really want to set a class on it, you can do this just after that:
ul.setAttribute('class', 'your_class');
Or whatever you prefer.
Alternatively, given that it already has an ID, you can add that ID to your existing CSS and thus style it directly. This would be best as it doesn’t involve changing the code, so if you upgrade the widget later, you don’t have to hack the code again. Just add “#delicious-list” to the given chunk of CSS code that does the icon thing.