“Get up close and personal! The amount of time you use the butt of your gun to inflict a bloody nose (or worse) at close quarters reflects how important you deem this perk to be. But it does increase the flexibility of more complex combat situations where you may not wish to switch between a favoured firearm and a different weapon type (like a punching weapon).”— Fallout 4 Vault Dweller's Survival Guide description
Gun bashing does double damage and has an increased chance to cripple your opponent. It may also inflict a Critical Hit.
Gun bashing will not deal as much damage as melee or unarmed weapons, especially when the latter have been modded or paired with Iron Fist or Big Leagues. This is because:
The "best" gun bashing weapons are essentially limited to common-place mods such as the Large Bayonet muzzle mod; so there is a hard early-game cap on itemization for gun bashing.
Gun bashing does not benefit from Big Leagues itself despite being a melee attack.
Gun bashing also does not benefit from any of the characteristics of the gun itself (such as bonuses from Commando or legendary weapon effects).
That being said, gun bashing is still very useful; it is virtually guaranteed to stagger enemies, so it can buy valuable time against enemies who have gotten too close for comfort. By mixing up gun bashes with normal weapon shots, one can keep an enemy staggered while also conserving ammo by doing some damage for free (albeit at a small AP cost). Moreover, gun bashing can be used to artificially shorten the reload time of a weapon: while naively gun bashing while reloading will cancel the reload, gun bashing after a new magazine has been inserted into the gun will let you perform a quick melee attack and then have a fully loaded gun upon completion of the melee attack.
In addition, starting with rank 2 of Basher, being able to potentially cripple your opponent with a gun bash can be very helpful, especially if you cripple legs to allow yourself a retreat.
Rank 2 gives an 11% chance to cripple the limb you strike. Rank 3 increases this to a 16% chance.
Rank 4 gives a 15% chance to cause bonus critical damage.
Bashing damage is affected by Strength.
Bashing damage is affected by enemy ballistic armor. Armor-piercing weapon mods, the legendary Piercing prefix, and the Rifleman perk actually do affect bashing with those weapons.
Bashing critical damage is affected by anything that increases critical damage. The Better Criticals perk and any weapon mods (such as the Lucky legendary prefix or a calibrated receiver) that increase critical damage will affect bashing criticals with that weapon.
The Overdrive chem will stack with the Basher perk for 40% total bashing critical chance.