Asylum Visitor's triggered ability checks the active player's hand as the upkeep begins and as the trigger resolves. If that player has a card in hand as it resolves, you won't draw a card or lose 1 life. Notably, if you control multiple Asylum Visitors during your upkeep, whichever one's first ability resolves first will stop the other's first ability from having any effect unless you have a way to get the card you drew out of your hand before it resolves.