When you throw a basketball, you give it a boost of energy for it to fly through the air. If the basketball isn't totally inflated, then more of that energy gets spent deforming the ball, that is, squishing it ever so slightly so it's no longer a perfect sphere. That energy spent on changing the shape of the ball no longer contributes to the "kinetic energy" of the ball, which is the energy of its motion through the air. As a result, if you throw an inflated ball and a deflated ball in exactly the same way, the inflated basketball will pick up more speed than the deflated one and thus travel farther.

The second part of this answer gets a little more complicated. Once the basketball leaves your hands and soars through the air, it's probably spinning backwards and experiencing air resistance. I say it's spinning backwards because the standard shooting form where one hand aims the ball and the other hand "flips forward" to launch it will give the ball backspin. A ball traveling through the air with backspin experiences something called the Magnus effect, which gives it a bit of lift - it will fly higher than it would if it were either in a perfect vacuum or not spinning. This Magnus effect happens because the spinning ball changes the direction of the movement of the air flowing around the ball. If the ball flies higher, it will go farther before it hits the ground. If the ball were deflated, it wouldn't be quite a perfect sphere, and this Magnus effect gets disrupted.

There's quite a bit more that goes into this answer, since maybe a basketball with only a little bit less air than the standard pressure could be easier to grip, or maybe a different way of throwing the basketball could deform the ball into more of a football shape that's more aerodynamic. Based on these two effects, however, I would say that the higher the pressure inside the basketball, the farther you can throw it.

Pumping more air into a basketball will not necessarily change the distance you can shoot it by a significant amount. The main effect of pumping more air into a basketball will be the amount it bounces when you are dribbling or after you shoot and it hits the backboard or the rim.

Pumping more air into the ball will increase the air pressure, which is a force that acts against the ground, backboard, or rim when the ball hits one of those objects. This force is what increases the 'bounciness' of the basketball. Unfortunately, pumping the ball with more air won't help you shoot from farther away, because the air pressure only helps the ball bounce farther after contacting an object. In fact, more air pressure might make it harder to shoot because the basketball will bounce off the rim harder, making it less likely to go in!

Let's start to address this question using the following hypothesis:
If I throw two differently-inflated basketballs with the exact same force from my arm, then the basketball with more air pumped into it will fly farther than the basketball with less air pumped into it. What would make this true? What affects the distance of a throw?

From Newton's first law, we know that
F = m a,
or Force = Mass * Acceleration.

If we know the Force of the throw is the same in both situations, and we know the Mass of the ball is the same in both situations (the rubber of the ball has much more mass than the air inside it; therefore the mass of the ball is basically the same no matter how much air is in inside it), then the Acceleration of the ball (how much it moves in response to the force from the throw) is the same no matter how much air is inside it. But what if the throw isn't the only force acting on the ball? We know there's (1) gravity, but that is true regardless of how inflated the ball is, and (2) air resistance.

Air resistance is an interesting one because it has to do with the interaction of air molecules with the surface of the ball itself: the ball must displace air in order to move through it. The motion of the ball through the air generates friction at the surface, which creates a drag force opposing the ball's motion. The spherical shape of the ball is not particularly aerodynamic (drag-resistant), but a more inflated ball has higher tension on the rubber of the surface, which is more aerodynamic because it creates less friction with the passing air. That means a more inflated ball would feel less drag and fly farther than a less inflated ball thrown with the same force, which would confirm our hypothesis.