Device independent dimension witnesses provide a lower bound on the dimensionality of classical and quantum systems in a "black box" scenario where only correlations between preparations, measurements and outcomes are considered. We address the problem of the robustness of dimension witnesses, namely that to witness the dimension of a system or to discriminate between its quantum or classical nature, even in the presence of loss. We consider the case when shared randomness is allowed between preparations and measurements and we provide a threshold in the detection efficiency such that dimension witnessing can still be performed.