How does smoking affect your blood and/or veins?

Several ways. Smoking causes constriction of the blood vessels. This occurs less commonly in the veins as compared to the arteries. Smoking is one of the major risk factors for the development of atherosclerosis (hardening and narrowing of the arteries). In addition to this it can lead to intrinsic lung disease, which could then lead to thickening of the blood(polycythemia) which could lead to blood clots.