In the recent years, dust storms (DSs) pose a serious threat to critical infrastructure such as power distribution networks (PDNs). During DSs, the contamination of insulators, increases the possibility of damage to the PDNs insulation system and flashover induced power outage may occur. Power outages disrupt the performance of other urban infrastructures and, in addition to heavy financial losses, cause public dissatisfaction. Although this issue is of particular importance in areas with humid climate, a few studies have been reported on PDNs resilience improvement against DSs. This paper proposes a novel cost-based optimization model to make PDNs more resilient to DSs considering uncertainties. The proposed model is based on the two-stage stochastic mixed-integer programming (SMIP). In the first stage, decisions are made to equip repair crews (RCs) with insulator washing machines, hardening distribution lines with silicone-rubber insulators (SIs), and deploy backup distributed generators (DGs). Decisions in the second stage include network reconfiguration, RCs routing, DGs power dispatch, and load shedding as the critical options for PDN outage management during/after DSs. Case studies are evaluated in the IEEE 69-bus test system and a real 209-bus PDN in Khuzestan province, a coastal province in southwestern Iran. The simulation results at different budget levels have confirmed the efficiency of the proposed model for cost-optimal resilience enhancement planning of PDNs against DSs.