Blightcaster's ability is a triggered ability that triggers whenever you (the controller) casts an enchantment.
Claustrophobia doesn't say anything about preventing active or triggered abilities, it just taps the creature.
If in doubt, expect no more than what the cards say.
The ability triggers. Blightcaster's ability is a triggered ability, which means it doesn't matter if it is tapped or not.
Yes, the ability will still work.
Under very old and outdated rules of MTG, triggered abilities of artifacts didn't work if the artifacts were tapped. These rules were changed in the 6th edition though, so now being tapped or untapped has no effect on a triggered ability, unless that ability specifically says "if [card name] is untapped...".
Also, as Blightcaster is a creature and not an artifact, even under the very old rules being tapped wouldn't matter here.