When people rush over the soul sand they normally fall a few times, but if you take your time you shouldn't fall at all. Once you get on the first 2 block platform you'll want to get as close to the edge as possible before sprint jumping to the first single block. From the first single block you'll want to back up to the back half the the platform & sprint jump to the 2nd single block. For the rest of the jumps just get as close to the edge as you can before sprint jumping to the next platform.
For the sponge jumps you'll want to get as close to the edge again as you can. When you sprint jump this time you'll want to continue running forward & swimming upon landing. You're using your momentum to basically bypass the first sponge platform to land on the 2nd. To transfer from the sponge to the netherrack you just swim off the corner closest to nettherrack.
The fire jumps can be timed so the damage doesn't affect your jumps. However...I never do that...I just go full speed. Sometimes I make it, sometimes the fire damage messes up my jump & makes me fall. Either way...it doesn't take too long to get past.
Once you make it up that ladder if you're low on health you need to hurry up & hit the pressure plate to save your progress or get in the pool of water.