Mike,
If you really want to, you can easily remove it.
Start at one of the front corners and cut off about 4 to 6 inches at a time. Then tack weld the pieces back together. Don't get in too much of a hurry welding it back together or you'll create warping. If you do about 4 to 6 inches at a time, the pieces will remain together and make it easier to weld.
If you watch shows like Overhaulin' or Trucks, they've done this before. I've recently done it on my '57 cab with no problems. Make sure that you don't cut out too much at a time, or cut too close to the roof. Leave a little metal to weld back together.
The cab looks much 'smoother' without the drip rail. I'm not sure how much water will run in the truck during a downpour as I'm a long ways away from reassembling my truck. Hopefully, new weatherstripping will take care of any leaks!
Lane