A finger-mounted electronic device may include a body that serves as a support structure for components such as force sensors, accelerometers, and other sensors and for haptic output devices. The body may have first and second side body members that leave the finger pad exposed and an upper body member extending between the first and second side body members. Some or all of the body may be covered in fabric or leather. Fabric may wrap around the first and second side body members and may extend across the upper body member. The fabric may cover electronic components. A touch sensor may have electrodes that are formed from conductive material on the fabric or conductive strands in the fabric. Infrared-reflective ink may form visual markers on the fabric for an infrared tracking system. The fabric may have light-transmissive portions that overlap optical components.