I don't see any reason to stop giving gifts to your godchildren or any other family members. Why would you stop giving someone a gift just because they turned a certain age? Everyone in my family gives gifts to everyone; parents, siblings, aunts & uncles, cousins, nieces & nephews, godchildren, grandchildren, etc.
My godparents are my aunt and her ex-husband. My aunt always gives me something for Christmas and my birthday (usually a gift card now that I'm older). My godfather lives in France and I haven't seen him since I was a toddler, when he divorced my aunt and fled to England to avoid paying child support to his four children. He's in Florida right now with my cousin who just had a baby, so he has somewhat made up with at least one of his children, but doesn't appear to have any interest in me.