The stripes were created by a special device that was fitted to the nozzle end of the tube: a tube within the tube, if you will, about one inch in length and perforated with a ring of small holes around the top.
Toothpaste tubes are normally filled from the flat end, which is then folded over and sealed. In the case of Stripe, a red toothpaste was first filled around the special fitting; the white toothpaste, filled second, held the red toothpaste in place at the top of the tube.
When the tube was squeezed, the white toothpaste would run through the special inner tube, while the pressure of the squeeze simultaneously forced the red toothpaste through the tiny orifices at the end. With the flow of red matched to the flow of white, the toothpaste emerged from the nozzle perfectly striped.
Via Straight Dope