If the standard hydrogen electrode acts as a cathode, the reaction at cathode will be:
H+ (aq) + e-→ H2 (g)
2H+ (aq) + 2e-→ H2 (g)
H2 (g) → 2H+ (aq) + 2e-
H2 (g) → H+ (aq) + e-
Reduction reaction takes place at cathode. So, a balanced reduction half reaction will be the reaction representation at cathode.
If the standard hydrogen electrode acts as an anode, the reaction at anode will be: