I HIGHLY recommend Keegan's The Face of Battle, particularly the chapter on Agincourt. He talks at great length and in great depth about the use of the bow on the battlefield. While it's a different battle, a different war (400 years after Hastings), and a different type of bow, it nonetheless is immensely helpful in imagining the impact ranged weapons might have had on a medieval battlefield.

In specific terms of Hastings, it's a bit hard to calculate the impact that bows had upon the battle. The English side had only a smattering of bowmen, while about a quarter of the Norman army were bowmen, so we can at least say that the victors had more bowmen. Additionally, Harold II, leader of the losing side's army, might have been killed by an arrow in the eye (this is a controversial topic, however, so I urge caution).

On the other hand, there's some indication that the Norman archers weren't particularly decisive, either. They were firing uphill, and most of their arrows either bounced off the enemy's shields or flew harmlessly over their heads. Furthermore, the Norman archers were ironically hindered by the lack of arrows being loosed back at them, which meant that no arrows were available once their supplies ran out. When the English responded with ranged attacks of their own, they mostly consisted of axes, javelins, and stones, rather than arrows.