Let's go step by step. First of all, what does it mean to cast a spell?
701.4a To cast a spell is to take it from the zone it’s in (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect. A player may cast a spell if he or she has priority. See rule 601, "Casting Spells."
So, as soon as you put your spell on the stack, even before it resolves, it is already being cast. The ability on the Viashino triggers:
603.1. Triggered abilities have a trigger condition and an effect. They are written as "[Trigger condition], [effect]," and begin with
the word "when," "whenever," or "at." They can also be expressed as
"[When/Whenever/At] [trigger event], [effect]."
603.2. Whenever a game event or game state matches a triggered ability's trigger event, that ability automatically triggers. The
ability doesn't do anything at this point.
603.3. Once an ability has triggered, its controller puts it on the stack as an object that's not a card the next time a player would
receive priority. See rule 116, "Timing and Priority." The ability
becomes the topmost object on the stack. It has the text of the
ability that created it, and no other characteristics. It remains on
the stack until it's countered, it resolves, a rule causes it to be
removed from the stack, or an effect moves it elsewhere.
So, the ability is put on the stack above the enchantment, so in a normal scenario the creature would get the +3/+3 even before the enchantment resolves!
Even so, the triggered ability is still an ability. It goes on the stack and can be responded to like most abilities and spells. Unless your opponent lets the triggered ability resolve before putting his Dismember on the stack, Dismember would kill your creature.