The paper deals with an axially moving viscoelastic structure modeled as an Euler-Bernoulli beam. The aim is to suppress the transversal displacement (transversal vibrations) that occur during the axial motion of the beam. It is assumed that the beam is moving with a constant axial speed and it is subject to a nonlinear force at the right boundary. We prove that when the axial speed of the beam is smaller than a critical value, the dissipation produced by the viscoelastic material is sufficient to suppress the transversal vibrations. It is shown that the rate of decay of the energy depends on the kernel which arise in the viscoelastic term. We consider a general kernel and notice that solutions cannot decay faster than the kernel.