The damage done per hit is twice normal for the weapon used per four experience levels of the thief,
i.e. double damage at levels 1-4,
triple at 5-8,
quadruple at levels 9-12, and
quintuple at levels 13-16.
Note that striking by surprise from behind also increases the hit probability by 20% (+4 on the thief's "to hit" die roll). (DMG)

Q: Can thieves back-stab or assassins
assassinate with missile weapons?
A: Thieves cannot use missile weapons for
back-stabbing attacks, but assassins apparently
can do so (going by a strict rules
interpretation), though this is not recommended.
(139.67)

Q. Does a thief's backstabbing bonus
apply to missile fire from behind?
A. No.  The attack is treated in the same
manner as a rear missile attack by
anyone else.  A thief would not cause
double damage with a missile attack
from behind.  The backstabbing attack
of thieves can only be carried out
with melee weapons, and represents
a thief's ability to sneak up behind a
potential victim and strike for maximum
effect.
(Imagine #19)

Q: If a thief using two weapons
makes a back attack, does he get his
"to hit" and damage modifiers for
both weapons?
A: The + 4 "to hit" and the damage multiplier
only apply to the first blow; the second
weapon gets the +2 "to hit" modifier for a
rear attack but no damage modifier. The
same holds true if the thief gets multiple
attacks due to surprise.
(139.67)

Q: When a thief makes a back attack,
are any damage bonuses for
strength or a magical weapon also
multiplied?
A: No. The multiplier applies only to the
weapon's base damage; other damage
bonuses are applied after the multiplication
is made.
(139.67)

Q: Can a thief ever get more than
quintuple damage for back-stabs?
A: No. Quintuple damage is the limit.
(139.67)

Dispel Confusion: Backstab +

ADQ: When a thief successfully sneaks up
to a victim and backstabs, should there
not also be a surprise roll? As the thief
truly did surprise the opponent, and
other character classes in this situation
would have the chance to attack in each
surprise segment gained, why not the thief?
or is maximum surprise assumed
-- modified by dexterty?
ADA: The degree of surprise is not assumed.
At the 1st confrontation, the thief
gains the backstab bonuses (+4 "to hit"
& multiplied damage) only if a standard
roll indicates surprise. If the victim is not
surprised, only the +2 bonus "to hit" from
behind applies to the backstab attempt, and
multiple damage is disallowed, since it is
contingent on surprise. Assuming surprise,
multiple attacks may be possible (as per
normal surprise rules); if so, the +4 "to
hit" bonus applies for all such attacks, but
only the 1st gains the multiplier, as the
victim who has been hit once does not remain
surprised per se. Another backstab
attempt against the same victim may be
made by a different thief, or by the same
thief if he or she disengages, departs, and
successfully gains surprise upon returning.
Surprise is not possible if the victim is
aware of the thief's presence and position.
(Polyhedron #31)

TheDungeonDelver wrote:
Gary: Backstab rule in 1e - meant to be melee only, or can it work ranged (e.g., "sniper fire")?

Not for a Thief--hand-weapon strike only in such case.
For an Assassin i would allow it.

Back Stabbing: Opponents aware of the thief will be able to negate the attack form.
Certain creatures (otyughs, slimes, molds, etc.) either negate surprise or have no definable "back",
thus negating this ability.