How can I run cat6 from my basement to the second floor?
You're almost certainly going to hit the top plate of the 1st floor wall, and possibly some fire blocking and duct bracing, so doing this without opening up the first floor wall is going to be difficult if not impossible. Not only do you have insulation to worry about, but there's also the vapor barrier that you don't want to tear by forcing the fish tape down.

Best option is to relocate to an interior wall, preferably one near a closet on the first floor so you can install some access panels high and low inside the closet should you ever need to fish more wiring.

Next best option would be to run directly through wall to the outside of the house and down some conduit from there. Make sure to seal the wall back up so it's watertight.