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The aim of this paper is to obtain an estimation of Hausdorff as well asfractal dimensions of random attractors for a class of stochastic partialdifferential equations with delay. The stochastic equation is first transformedinto a delayed random partial differential equation by means of a randomconjugation, which is then recast into an auxiliary Hilbert space. For theobtained equation, it is firstly proved that it generates a random dynamicalsystem (RDS) in the auxiliary Hilbert space. Then it is shown that the equationpossesses random attractors by a uniform estimate of the solution and theasymptotic compactness of the generated RDS. After establishing the variationalequation in the auxiliary Hilbert space and the $\mathbb{P}$ almost surelydifferentiable properties of the RDS, an upper estimate of both Hausdorff andfractal dimensions of the random attractors are obtained.

Topological dimensions of random attractors for stochastic partial differential equations with delay