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If a 6th level Sorcerer with Elemental Affinity (lightning) casts Witch Bolt, is only the inital bolt's damage is increased, or the damage of every one of them?

Elemental Affinity:

[...] when you cast a spell that deals damage of the type associated with your draconic ancestry, add your Charisma modifier to that damage.

Witch Bolt:

[...] on each of your turns for the duration, you can use your action to deal 1d12 lightning damage to the target automatically.

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Only Once

Elemental Affinity adds damage "when you cast a spell..."

The damage of witch bolt that can be applied on subsequent turns is not "casting a spell" but instead taking a special action granted to you by the witch bolt spell. Since you're not casting a spell, the additional damage from Elemental Affinity will not apply.

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Yes, only to when you first cast the spell.

Per The Player's Handbook Errata,

Elemental Affinity (p. 102). The damage bonus applies to one damage roll of a spell, not multiple rolls.

Old Answer

Since you're not casting a spell to automatically deal the damage, Elemental Affinity only applies for the initial damage.

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