The aim of this paper is to obtain an estimation of Hausdorff as well as fractal dimensions of random attractors for a class of stochastic partial differential equations with delay. The stochastic equation is first transformed into a delayed random partial differential equation by means of a random conjugation, which is then recast into an auxiliary Hilbert space. For the obtained equation, it is firstly proved that it generates a random dynamical system (RDS) in the auxiliary Hilbert space. Then it is shown that the equation possesses random attractors by a uniform estimate of the solution and the asymptotic compactness of the generated RDS. After establishing the variational equation in the auxiliary Hilbert space and the $\mathbb{P}$ almost surely differentiable properties of the RDS, an upper estimate of both Hausdorff and fractal dimensions of the random attractors are obtained.