I was actually thinking about this a lot recently, and a post at Fighting Fantasist covered it today. In general, I love 1d6 damage for most weapons.
The only exception would be polearms and two-handed swords. If a player chooses to use one of these weapons, he will gain no benefit over just using a normal sword. In addition, he will not be able to use a shield.
In S&W White Box, a shield gives +1 Armor Class. This doesn't sound like much to modern D&D players. However, in White Box the Dexterity AC bonus is optional. And if it is used, the most benefit you will gain is a +1 AC. This is a far cry from the +4 AC you see in 3e (not sure what 4e does). So that +1 AC bonus from the shield is pretty damned important.
So, if a player really wants to wield a two-handed sword, he will be taking a penalty to his combat ability, as compared to another fighter who takes a weapon and shield.
My house rule for this is that polearms and two-handed swords do d6+1 damage. This makes them deal out slightly better damage, in compensation for the loss of a shield. It makes two handed weapons more attractive, but with the trade-off of the wielder's AC being worse.