Ok, four points here. For the most part, it really depends on your game and if your players are ok with punishing adventures. D&D is not like systems like 40k, where its ok to kill a player character outright with one good shot.
Damage Balance (spells by themselves)
As for increasing the raw power of a spell, it really doesn't do it that much over time. In this sense I'd call it balanced.
Consider a 3rd level Fireball, where an 11 is needed to save. I chose this spell, because for its level Wizards of the Coast has admitted to overtuning the damage to make it interesting.
Before:
The average damage done on a successful save is given by
\$ \textrm{Avg}_S = (4 \cdot 3.5) = 14\$
The average damage done on a failed save is given by
\$ \textrm{Avg}_F = (8 \cdot 3.5) = 28\$
Thus the average damage is given by
\$ \textrm{Avg} = \frac{1}{2} \cdot \left(14 + 28 \right) = 21 \$
If you included a 1/20 crit rule then
The average critical damage is given by
\$ \textrm{Avg}_C = (16 \cdot 3.5) = 56 \$
The average damage on failed and succesful saves remains the same, but the probability of a failed (non-critical) save decreases from \$\frac{10}{20}\$ to \$\frac{9}{20}\$.
Thus the new average damage is given by
\$ \textrm{Avg} = \frac{1}{20}\cdot(56) + \frac{9}{20}\cdot(28) + \frac{10}{20}\cdot(14) = 22.4\$
As you can see you only increased the average fireball damage by about 1.4 per target.
One-Shot Balance
If you consider the impact of a failure for characters at 5th level, a crit failure could potentially mean instant death... whereas a regular failure would mean you might just drop them to 0 HP.
Failing:
\$ 8\mathbf{d}6 = 3.5 \cdot 8 = 28\$
Crit Failing is essentially twice that:
\$56\$
A 14 con wizard who rolls average health will have: 30 health. IF they roll average. For point buy, a fireball is already in danger of dropping them to 0 HP. Twice that could potentially insta-kill that character. (If damage reduces you to 0 HP and the remaining damage exceeds the character's max HP, you die instantly, without making death saving throws.)
In this sense I'd also call it unbalanced - at least at the level certain spells are introduced. It could also lead to easier TPKs because your party's healer accidentally died instantaneously at the beginning of a fight and your party wasn't expecting it.
Save Balance
Does it make sense to double damage someone with a potential 16d6 fireball if, say, their Dex save is a base of 8 and the spell save DC is only 16?
In this sense I'd call it unbalanced.
If they rolled a total higher than half the spell save DC - Why should I suffer the same damage rolling a 9 (because I've got a 20 Dex and am proficient) where another character could roll a 2 (not proficient with 12 Dex). At worst, if I were to implement something like this in my games, your total on the roll would need to be less than half of the spell save DC before an auto 1 could crit. Would give the death feel you're looking for while not hosing those who are proficient.
Thematics
Does it even make sense to crit someone with certain spells?
Personally, I'd say no. The spell damage is supposed to cover how much more or less you got clobbered by a spell.
A few important interactions
Finally, what about paralyzed creatures? (They normally auto-fail Dex Saves.)
Should they get the double crit damage too, where archers would not? Spell interactions alone are the one main reason I wouldn't allow this, as things like Stunning Strike and Hold Person/Monster would now hold all the weight in the world, when traditionally they only really help melee. Sorcerers' Metamagic especially would and could become extremely potent; evocation wizards' maximizing damage from Overchannel, and divination wizards' Portents become brutal (especially combined with things like the Disintegrate spell).
Some alternative options if you want nat. 1s rolled for saves to feel special
If the character is resistant to the damage, remove the resistance for that casting of the spell. (Doesn't alter the damage of the spell, just the amount the character expects to take.)
You could cause the damage roll to reroll 1s for free as per the
feat in the PH that does the same. (This actually helps the damage
lot.)
Require the total of the roll to be less than half the target save DC.
Attach a sort of stun to the character or dazed. Whereby their next offensive round of offensive abilities with the next minute either has disadvantage or creatures gain advantage on the next save for spells cast by this character (makes it even for casters and doesn't discriminate on type of spell). (Losing action economy on a damaging thing can at least make the 1 painful while not totally boning the player, and if the situation is dire they can risk resources on spells to save the day.)