The problem of stability of a viscoelastic anisotropic fiber reinforced plate under the action of a rapidly increasing (dynamic) shear load in a geometrically nonlinear formulation is considered. The mathematical model of the problem is described by a system of nonlinear partial integro-differential equations with singular relaxation kernels. The Bubnov–Galerkin method is used to obtain systems of ordinary nonlinear integro-differential equations. The solution of the system of resolving equations is carried out by a numerical method based on quadrature formulas. To substantiate the accuracy and adequacy of the obtained results, a test problem is solved. A stability criterion for reinforced plates under the action of shear loads is introduced. Depending on various geometric, physical, and mechanical characteristics of the material, the behavior of the reinforced plate is investigated. In particular, it is shown that taking into account the viscoelastic properties of the material leads to a decrease in the critical time, and therefore in the critical force. Depending on various geometric and physical parameters, the difference in critical time values for elastic and viscoelastic plates in some cases is more than 15 %. It is also shown that an increase in the angle of fiber direction in the plates leads to a decrease in the critical time. Among the single-layer reinforced plates, the plate with a fiber direction of 0° is the most resistant to shear loads. An increase in the number of layers in a reinforced plate while maintaining its thickness does not always favorably affect the stability of the plate. In the case of three-layer viscoelastic plates made from KAST-V material with fibers oriented in the direction of 45°/–45°/45°, they are less stable than double-layer plates but more stable than single-layer ones while maintaining equal thicknesses of all three structures.