When using a weapon without reach, does the range of booming blade still double if you have spell sniper?
As of November 10th 2020, this doesn’t work.
On November 10th, 2020, booming blade was changed to have a range of self, so is no longer eligible for spell sniper. See this errata document for the Sword Coast Adventurer's Guide.
I will preserve the original answer for those who elect to ignore this errata.
The range of the spell doubles, but trivially so.
The first bullet of Spell Sniper says:
When you cast a spell that requires you to make an attack roll, the spell’s range is doubled.
This answer goes into detail about the interaction between Spell Sniper and booming blade. The short verion is, as Rubiksmoose writes:
Spell sniper will work because booming blade has an attack roll involved in it as long as you have a weapon capable of making a melee attack.
So it technically doubles the range of the spell, but this adjustment is entirely trivial without a reach weapon: with a reach of 5 feet, you can still only make an attack on a creature within 5 feet. Booming blade doesn’t extend the reach of your non-reach weapon, but the range of the spell is technically 10 feet with spell sniper, though it still fails if you do not make a melee attack against a creature (or cannot).
Unless you are a bugbear!
The bugbear playable race has a feature called Long-Limbed:
When you make a melee attack on your turn, your reach for it is 5 feet greater than normal.
So even with a non-reach weapon, Spell Sniper would enable the bugbear to use booming blade on targets 10 feet away. In this case, it's actually very important to understand that the range of the spell is actually doubled.
Booming Blade is limited by the weapon's reach
Booming Blade states:
As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell’s range, otherwise the spell fails.
The melee attack must target a creature within the range of the spell but that does not affect the reach of the melee weapon. If the target is outside of your reach then the spell fails.