We perform a search for beyond-the-Standard-Model (BSM) dimension-six operators relevant to the Higgs boson at the Large Hadron Electron Collider (LHeC) and the Future Circular Hadron Electron Collider (FCC-he). With a large amount of data (few ab−1) and collisions at the TeV scale, both LHeC and FCC-he provide excellent opportunities to search for the BSM effects. The study is done through the process e−p→hjνe , where the Higgs boson decays into a pair of b¯b, and we consider the main sources of background processes, including a realistic simulation of detector effects. For the FCC-he case, in some signal scenarios, to obtain an efficient event reconstruction and to have a good background rejection, jet substructure techniques are employed to reconstruct the boosted Higgs boson in the final state. In order to assess the sensitivity to the dimension-six operators, a shape analysis on the differential cross sections is performed. Stringent bounds are found on the Wilson coefficients of dimension-six operators with the integrated luminosities of 1  ab−1 and 10  ab−1 , which in some cases show improvements with respect to the high-luminosity LHC results.