The cold-water pipe (CWP) used for Ocean Thermal Energy Conversion (OTEC) floating plant inevitably faces instability and failure due to mass seawater transportation and severe operating conditions. This paper investigates the stability of the pipe under different boundary conditions to improve the safety of CWP operation. Utilizing the delta function and Heaviside function, the control equation for CWP vibration with clump weights is established based on the Euler-Bernoulli theory. The damping force of the viscoelastic dissipation should be considered in the theoretical model but has been previously ignored in past studies. The Generalized Integral Transform Technique (GITT) is employed to solve the vibration control equation, and the newly developed model is validated by comparing it with previously published data. Numerical results indicate the significant role of viscoelastic damping forces, and neglecting this damping force can lead to inaccurate findings regarding the effects of the clump weight. The viscoelastic dissipation, mass ratio, the clump weights and positions significantly affect the critical flow velocity, frequency, and modal shapes. Our research suggests that pipe stability can be enhanced by adjusting the mass ratio, the clump weights and positions depending on specific boundary conditions. These conclusions offer valuable guidance for practical engineering applications.