Broken capillaries is a problem with the tiny blood vessels that become visible near the surface of the skin and usually appear as streaks or blotches.They are commonly known as spider veins because of their tree branch, or spider web-like, shape.eople with fair skin are more prone to having broken capillaries, or telangiectasia, than people with a dark or olive complexion. People with dry and dehydrated skin, as well as very sensitive skin, have thinner and less protected layers of tissue, thus making them more susceptible to this problem.