Answer
The ratio is $\sqrt{2}$:
$$\frac{v_{10}}{v_{20}}=\sqrt{2}$$
Work Step by Step
If we keep the launch angle constant then the maximum height scales proportionally to $v_0^2$. This means that, since the maximum height is twice as big, the square of the magnitude of the initial velocity of the first projectile two times greater than that of the second projectile i.e.
$$v_{10}^2=2v_{20}^2\Rightarrow\frac{v_{10}^2}{v_{20}^2}=2\Rightarrow\frac{v_{10}}{v_{20}}=\sqrt{2}$$
