Spoon the yogurt onto several layers of heavy-duty paper towels; spread the yogurt to a 1/2 inch thickness. Cover with additional paper towels; let stand for 5 minutes to remove the excess moisture.
Position the knife blade in a food processor bowl; scrape yogurt into the bowl, using a rubber spatula. Add the goat cheese and olive oil; process until smooth, stopping once to scrape down the sides. Add the spinach, basil, and red pepper; process until the spinach is chopped.
Spoon the mixture into a bowl; cover and chill at least 2 hours. Serve with assorted crackers.