Any process that modulates the frequency, rate or extent of synapse structural plasticity. Synapse structural plasticity is a type of cytoskeletal remodeling; this remodeling is induced by stimuli that can lead to long term potentiation and it can be activity-dependent or -independent. Examples of cytoskeletal changes include the formation of new spines and increase in spine size; this can be accompanied by the insertion of greater numbers of glutamate (or other neurotransmitter) receptors into the post-synaptic membrane.