Beat cream cheese with half-and-half with an electric mixer in a bowl until smooth and easy to spread; stir in chives.
Use a large round cookie or biscuit cutter to cut 24 rounds out of the bread slices. Spread each round with cream cheese mixture and top each canape with 1 or 2 thin cucumber slices and a basil leaf. If making ahead, cover tea sandwiches lightly with plastic wrap and chill.