The Basher perk improves the effectiveness of gun bashing by adding extra damage, and possibly crippling a limb or causing a Critical Hit.”— Fallout 4 loading screen hints
The Basher perk is centered around improved the melee attack with firearms and has four ranks. With the first rank, gun bashing does an extra 25% damage. The second rank raises this bonus bashing damage with 50% and allows bashes to sometimes cripple opponents. The third rank raises the bonus damage to an extra 75% and raises the chances to cripple an opponent. The fourth and final rank raises the bonus damage to an extra 100%, further increases the chance to cripple an opponent and may also inflict a critical hit.