Str is mainly needed for heroes who are required to initiate or to unleash their spells only when near an enemy. Giving high str to your hero only because of the dmg is a bad choice because a high stat gain can make a hero rather imbalanced, even if your base stat ain't that big. Before giving your hero any str stat you should first look at his EHP (Effective Hit Points).
EHP is armor/dodge/dmg reduction, if you combine the EHP with your HP you'll see the hero's true life and how well he'll do in battle, so it's recommended that you 1st calculate your agility and if you have any ability that grants you dodge/dmg reduction/+ armor, you should take them into account as well. Here's the tutorial on EHP:Damage and Armor - DotA Mechanics
I recommend checking it out