In 1st Edition Mongoose Traveller, you do not add your strength DM to melee damage.
It's incorporated into your roll to-hit, which indirectly affects damage.
On page 61 of the core rulebook:
The standard skill checks used in making an attack are:
Melee attack = 2d6 + Melee (appropriate specialty) + Strength or Dexterity DM (attacker's choice)
Subsequently, on page 65, under Damage:
Each weapon lists the damage it inflicts as a number of d6. Add the Effect of the attack roll to this damage.
The usual target number is 8+, so to hit someone you're wanting your total to be 8 or more (plus or minus whatever situational modifiers are going on).
The Effect here is the margin of success of the the attack (p50, Effect section). So for every point you get over 8 on your dice roll you inflict an extra point of damage. This means that high strength increases both your accuracy and maximum damage, but only if you choose to use your Strength DM for the attack roll.
In summary: When rolling to hit in melee, use the better of your Strength or Dexterity DMs. Don't apply Strength to the final damage total.