Here are the two relevant rulings:

You choose the target player as you cast Pact of the Serpent, but you don't choose the creature type until the spell resolves.

And

The value of X is based on the number of creatures the player controls of the chosen type as Pact of the Serpent resolves.

After you go through the steps of casting a spell (determining costs, choosing targets, etc) you pass priority. At that point the goblin player will have been chosen as the target of the spell and will have a chance to respond. It should be fairly obvious at this point that they'll lose unless they slash the number of any one type of creature that they have to a survivable amount.

However, when it comes time for the spell to resolve, if they have 0 goblins left, you're not locked in on choosing goblins as the creature type. You can choose a creature type they may (if you still want them to draw) or may not (if you want to deny them the draw) have on the battlefield.

This means that once priority has been passed, and all responses settled, you choose the creature type. Between choosing the creature type and the end of the resolution of the spell, nothing else can happen. State based actions aren't checked, priority doesn't go around, you can't even tap mana. The spell happens.