When calculating the length of an arc, does it matter which unit of measurement (radians or degrees) you use when measuring the central angle?

No, it does not matter.

If you measure the central angle in degress, you use the proportion

circumference / 360° = length of arc / central angle in degrees,

and if you measure the angle in radians, you use the proportion

circumference / (2π) = length of arc / central angle in radians.