In a medium saucepan stir together water, sugar, cranberries and ginger. Bring mixture to a boil, then reduce heat and let mixture simmer until sugar completely dissolves. Take care because cranberries will pop. Remove from heat, cover, and let steep for 15 minutes.
Strain ginger and cranberries from syrup mixture and let cool completely.
Assemble the bellinis by adding the sparkling wine to a champagne flute, filing halfway, top off with the cranberry-ginger simple syrup. Garnish with fresh rosemary, if desired.
Store any remaining simple syrup in the fridge. It will keep for about a week.