According to USA Today, Starbucks sells 4 million coffee drinks per day. According to the company's website, they serve 60 million customers each week, though it's important to note that Starbucks offers products and drinks other than coffee.
Starbucks also owns Seattle's Best Coffee, which serves, according to the company, "millions" of cups of coffee each day. So, there are two possible answers when determining how many cups of coffee Starbucks serves each day. There are the cups sold by the Starbucks stores throughout the world, and there are the cups sold by the parent Starbucks company, which owns two brands.