I would prefer to unlock adventure mode for non-seasonal characters using a seasonal character since I can make use of the shared stash I have already built up. I've tried Google, posting on one of the forums I frequent, and asking in General chat in game and nobody seems to know. Any help is appreciated!
Season and Non-seasonal are seperate and unlocking something in one won't do anything in the other. However if you unlock adventure mode in Seasonal, your character will go to non-seasonal when the season is over and adventure mode will be unlocked there as well.
Also, seasons do not have adventure mode locked at all, but you will have to complete the campaign to unlock adventure mode in non-seasonal.