First off, no, we don't see anyone closing the gate, but that doesn't necessarily mean that no one did so. It might have happened after the shot ended.
Even if we assume that the gate was left open, it doesn't necessarily mean that they chose to leave it open. Consider the circumstances under which the fence was erected.
The military cleared the surrounding areas, isolated a single neighborhood, and set up their defenses there.
"Cobalt" was put in place as a last resort, in case the crisis spiraled out of control, but the military probably believed that Cobalt would never be needed.
Thus, the military built the fences and gate under the assumption that they, the army, would always be in control of the gate.
This means that they thought there would always be a guard inside the gate, waiting to let patrols in and out of the compound.
This in turn means that they probably didn't put a control panel on the outside of the gate. Whoever was on guard duty would use the interior control panel to open and close the gate.
Furthermore, it makes more sense to have a single control panel inside the compound, because the whole idea of having a compound is to keep the people outside of it out. Giving the people outside a way to get in defeats the purpose of having a gate.
If this assertion is correct, there is a simple answer to the question of why they didn't close the gate: because they couldn't. The control panel was inside, so once they left, they didn't have access to it.
##But even so, this doesn't necessarily matter. Chances are, the gate was designed to close automatically after a few seconds. As such, there probably wasn't any need to close it manually.
