A key concept in the emerging field of spintronics is the gate voltage or electric field control of spin precession via the effective magnetic field generated by the Rashba spin-orbit interaction. Here, we demonstrate the generation and tuning of electric field induced Rashba spin-orbit interaction in InAs nanowires where a strong electric field is created by either a double gate or a solid electrolyte surrounding gate. In particular, the electrolyte gating enables 6-fold tuning of Rashba coefficient and nearly 3 orders of magnitude tuning of spin relaxation time within only 1 V of gate bias. Such a dramatic tuning of spin-orbit interaction in nanowires may have implications in nanowire-based spintronic devices.