Organic splinters and basically what you have inside your finger right now are pretty much the worse thing you can have in a healing wound. You'd be better off with a chunk of metal because as the basil breaks down inside your body, it'll provide a food source for the staph bacteria or whatever the hell else is on it. That bitch gets sealed in there and it's pretty much the perfect environment for that shiznit. Although I personally wouldn't recommend any doctor visits or topical anti-biotic (those motherfuckers love handing out anti-bios like they get off on doing it) unless you actually really need them (look for the insane redness) since broad and narrow band resistant bacteria aren't good for anyone.
With thorns and wood slivers they usually eventually get pushed out by the body naturally, but this is quite a bit different. What you could try (which is really good for tiny softwood splinters that can't be pulled out and might not get pushed out) is to mix some water and about half a teaspoon of baking soda into a sort of thick paste, put that shit on it, and cover it up. Hopefully in the morning or the next day (you might have to do it twice) that bitch will come out.
Also, since it's a pretty wide cut, could you not just submerge it in water and swish it around, hoping for the basil to just float off?
PS: Don't go to the doctor.
