Does Rhystic Study trigger if an opponent pays the one mana?

Rhystic Study reads: 'Whenever an opponent casts a spell, you may draw a card unless that player pays {1}.' The payment of {1} is a condition that prevents the triggered ability from having any effect — specifically, it prevents you from drawing. If the opponent pays, no card is drawn. The trigger itself does occur (the event of casting a spell happened), but the payment negates the draw outcome.

More precisely, the ability triggers whenever an opponent casts a spell (CR 603.1). As part of that trigger resolving, the opponent is given the option to pay {1}. If they pay, the 'may draw a card' effect is suppressed. If they do not pay, you may draw a card (CR 603.8 covers such 'unless' costs on triggered abilities).

It is important to understand that the trigger always goes on the stack when an opponent casts a spell — the opponent simply chooses whether to pay when the trigger resolves. The payment does not stop the trigger from being placed on the stack; it only stops the card-draw effect from happening.

Example: Your opponent casts Lightning Bolt. Rhystic Study triggers and goes on the stack. When the trigger resolves, your opponent pays {1}. You do not draw a card. If they had refused to pay, you could draw one card.

This is a common multiplayer source of confusion: paying the mana 'turns off' the draw, but the trigger still happened and the opponent still had to make that choice when it resolved (CR 603.1, 603.8).