Does running slowly worsen varicose veins on your legs?

No: The speed of running does not make a difference. Any exercise will help the venous circulation of the legs regardless of the speed.

Yes, running worsens: Running can damage weak one way valves within leg veins, worsening varicose veins. It is not how fast you run but how hard you slam your foot down. This generates tremendous torque on the valves, leading to damage and failure, the veins subsequently dilate. Walking is much more beneficial.

Not the speed: As drs. Farhy and presant said, pounding, more than speed, might have an influence on your veins and make them worse. Low impact exercise (biking, walking, swimming, etc.) is likely better. The most important thing though is that you exercise, regardless of what you do.