Constitution, at the very least in the GBA series of Sealed Sword/Binding Blade, Blazing Sword and Sacred Stones, affects two things: The ability to wield tomes and weaponry without a speed penalty and to determine if a unit can be Rescued by another unit (This does not apply to the Rescue or Warp staff.)
So, depending on how you want to lay out your strategies, you might want to give just enough constitution to units with very low Constitution so that they can wield certain weapons without the speed penalty. The penalty is determined by the Weight of the weapon or tome, and if the difference of x in x = CON - WT is negative, then the unit loses that much Speed in the battle calculations.
In Sacred Stones, Lute is a good example of a unit that benefits from some, but not much, Constitution. With a base 3 Constitution, she suffers from a penalty from most tomes. Take, for example, if Lute were to wield the Fimbulvetr tome (WT 12) with her base Constitution as a Sage (CON 3 + 1 due to promotion bonus), she'd suffer a whopping -8 penalty to her Speed when attacking with that tome.
Therefore, if you want Lute to be mobilized around the map by say, Pegasus Knights, increasing her constitution just a little bit so that the penalty isn't that great or is totally negated is a good idea. On the back side, as a rather frail unit, she might not be able to be rescued by other units with less Constitution. Evaluate your choices in strategies and plan accordingly.