No matter the foot type, proper support for the arch and heel will help to minimize foot pain and risk for injuries during running.
A high arch requires support along the longitudinal arch (bottom of the foot) and likely requires lateral foot support. With a high arch, the foot tends to land laterally (outside edge) as your foot hits the ground. Often times this will lead to lateral foot pain. A shoe with lateral support will help to minimize the asymmetrical foot strike.
A low arch also requires support along the longitudinal arch because the foot itself does not have an adequate arch. As the foot hits the ground, it lands flat which leads to over-stretching of the plantar fascia (bottom of the foot). Often times plantar fasciitis is next. A solution can be as simple as an over-the-counter shoe insert with adequate arch support.