1. Concept used: escape to infinity

To send a body from the Earth's surface to infinity with zero speed at infinity, the minimum required kinetic energy must equal the increase in gravitational potential energy.

2. Gravitational potential energy at Earth's surface

The gravitational potential energy of a mass $m$ at distance $R_E$ from the center of Earth is

$U = -\frac{GM_E m}{R_E}$

At infinity,

$U_{\infty} = 0$

So the required increase in potential energy is

$\Delta U = 0 - \left(-\frac{GM_E m}{R_E}\right) = \frac{GM_E m}{R_E}$

Hence minimum kinetic energy needed is

$K = \frac{GM_E m}{R_E}$

3. Use relation between $g$ and Earth's parameters

We know

$g = \frac{GM_E}{R_E^2}$

So,

$GM_E = g R_E^2$

Substitute into the kinetic energy expression:

$K = \frac{(gR_E^2)m}{R_E} = mgR_E$

4. Match with options

Thus, the required kinetic energy is

$\boxed{mgR_E}$

So the correct option is:

C: $mgR_E$