High-speed filming and force measurements by Tonia Hsieh and Thomas Roberts broke the stride into three phases. The slap of the splayed foot against the surface generates upward force. The stroke that follows drives the foot down into the water, still producing lift, and creates an air-filled cavity around the leg. The recovery pulls the foot back out before the cavity closes, which matters because pulling a foot through collapsing water would cost more than the stride gains.
Fringes of skin along the toes increase surface area and are spread during the slap. Small basilisks manage the trick easily; the physics scales badly with mass, so juveniles run further than heavy adults, which tend to break through and swim after a short distance. Reported runs are on the order of several metres to twenty or so.
Surface tension plays effectively no part, which is the usual misconception. The behaviour is escape, not routine travel, and basilisks swim well and can remain submerged for extended periods when pressed.