Suaidi Yahya, mayor of Lhokseumawe, says it aims to save people's "morals and behaviours".
Leaflets have been sent out to government offices and residents to inform them about the regulation.
Aceh is the only Indonesian province that follows Sharia.
Under the new regulation, the mayor says that women passengers are only allowed to sit "side-saddle" because straddling the bike seat violates Islamic values.
"When you see a woman straddle, she looks like a man. But if she sits side-saddle, she looks like a woman," Suaidi said.
He added that passengers who sat side-saddle rarely fell off.