I always play survivor mode in Asse: if you are not intrested in pvp and just want to finish the quests there, try to stay away of banners (and don't use them), don't try to pvp unless there is no choice, and avoid quests that ask you to kill other players. Still, some quests will ask you to kill enemy NPCs, so try to sneak into the places and don't kill anything if there are enemy players close: get in and out as fast as posible.
If you are not interested in do the quest, and if it is posible, you also can use the option to finish the quest instantly.