The way that works for me
I have dealt with swollen knuckles due to BJJ and judo grip fighting since 1997 (and I wrestled in high school years before that). I've messed around with literally dozens of different ways to tape my fingers, and this is the one way that works best for me. Check it out, and let me know if this works for you!
Step 1: Prepare the Tape
First things first: if you're serious about training, you are going to want to make sure you have the right tape. I recommend Johnson & Johnson COACH Athletic Tape - Pack of 32 if you want to invest in a year's supply or so, or if you want to go in on it with a buddy. Otherwise, just shop around for an individual roll, but MAKE SURE it's cloth athletic tape, not some cheap imitation.
Now start by tearing the tape in half, lengthwise, as pictured. This tape will retain its midway division all the way down to the end of the roll of tape, thus ensuring many days of finger-taping use.
Step 2: Assessing the Injury
This is my swollen knuckle. Spider guard gripping is normally one of my favorite things to do, but without a proper taping job, this is more or less out of the question for me right now. I'm going to start by taping above and below the big, swollen knuckle (pictured) and give the finger the support it needs. Basically, the idea is to not allow quite as much of a range of motion - If I make a fist, there's a great deal of pain and further potential for injury. Making a tape barrier will help with that.
Step 3: Tape Above and Below the Knuckle
In this step, you're basically going to make two separate little casts for your finger. The casts won't be made of plaster, of course, but of that Johnson and Johnson tape (or whatever comparable cloth tape you use). After wrapping above and below the knuckle about 5 times around for each "cast", try to make a fist. If your fist can clench far enough for you to be in pain, add more tape until you can't quite make a full fist. It's important to leave a little bit of "breathing room" in between the two strips of tape so that you can still bend your finger safely (and functionally, because hey, the purpose of doing this is to still be able to roll).
Step 4: The Buddy System
This last step isn't always necessary, but it always makes the bond considerably stronger and more secure. I'm essentially taping the injured, already wrapped finger to the next door buddy finger. The "buddy system" works oh so well, especially in conjunction with the previous step of creating the "finger casts." With the buddy system step, it's okay to criss-cross the tape underneath the fingers, but try to avoid crossing it on top, as this will seriously inhibit your gripping ability. With the method pictured, you should be good to go!
Step 5: Roll!
Be careful and use common sense, of course, but you can definitely train around an injured finger quite effectively without completely changing your game. I have run a BJJ gym in Richmond, VA for more than 8 years now (Revolution BJJ- seriously, stop by sometime if you're in town!), and as such, I've been privy to lots and lots of advice on how to tape fingers. I've never seen this particular method used by anyone else, so I wanted to share it with you here so that maybe your fingers can be saved some grief, and you can train effectively through a relatively minor injury. Happy healing, and happy training, and I'll see you on the mats one day!
Update: the result
Here's a picture of my (formerly) swollen knuckle after a few months of taping it. That's right- in order for your fingers to actually heal, you are going to need to keep up properly taping them for months, not weeks or days, unless it's a much more superficial injury than mine typically are.
As you can see here, the swelling in the middle knuckle, middle finger is all but gone (no, my hands are normally just gnarly like that). Consistency with the taping is absolutely paramount here- don't skimp on a single day, because that's the day you'll reinjure it! Also, consider taping your injured finger for no-gi training as well. You'll thank me later when your hand works again.