Recent experiments have tuned the monolayer 1T^{'}-WTe_{2} to be superconducting by electrostatic gating. Here, we theoretically study the phonon-mediated superconductivity in monolayer 1T^{'}-WTe_{2} via charge doping. We reveal that the emergence of soft-mode phonons with specific momentum is crucial to give rise to the superconductivity in the electron-doping regime, whereas no such soft-mode phonons and no superconductivity emerge in the hole-doping regime. We also find a superconducting dome, which can be attributed to the change of Fermi surface nesting conditions with electron doping. By taking into account the experimentally established strong anisotropy of temperature-dependent upper critical field H_{c2} between the in-plane and out-of-plane directions, we show that the superconducting state probably has the unconventional equal-spin-triplet pairing in the A_{u} channel of the C_{2h} point group. Our studies provide a promising understanding to the doping dependent superconductivity and strong anisotropy of H_{c2} in monolayer 1T^{'}-WTe_{2}, and can be extended to understand the superconductivity in other gated transition metal dichalcogenides.