It's gonna get finicky if your STR is higher than your DEX, since a finesse weapon will automatically use the higher of the two. But if you're limiting it to personal use on a character with a higher DEX, this should do it. Throw it into a .user file and add it as an adjustment:
Code:
<thing id="pSTRtoFin" name="Add STR to DEX Finesse Weapon Damage" compset="InPlay">
<tag group="OthAdjCat" tag="Equipment"/>
<tag group="Helper" tag="NoIncr"/>
<eval phase="Final" priority="10000"><![CDATA[doneif (field[pIsOn].value = 0)
foreach pick in hero from BaseWep where "wProperty.Finesse"
eachpick.field[wDamAttr].value += hero.childfound[aSTR].field[aModBonus].value
nexteach]]></eval>
</thing>