Originally Posted by Schmorgy
You might want to relax your grip a bit too. Bass strings are't like guitar strings where the calluses form from having your fingers cut up a bunch. They're big enough that the pressure is more evenly distributed across the finger, so if you're hurtin, you're probably strangling the neck of the bass. It takes a LOT less pressure to fret a clean note than some people think, and gripping too hard is one of the hardest habits to break so it's best done early on.
I think this is key, and what mjr. softie said is also a factor in this.
Make sure your action is set up correctly. That means action not too high so you don't waste energy, but not too low so you get rattle all the time. Playing "softer" with your plucking hand will help you get the action ever so low. Notice that by playing soft, nothing is said about dynamics or style. With enough practice you will still have all the dynamic range you need, down to a whisper even.
To find out how much strength you need to fret a note, fret a note as you would normally and gradually try to apply less and less pressure until the note does not sound cleanly anymore. Then add a tiny bit of pressure so it does ring again and viola, this is your minimum level of finger pressure needed. Try to keep it as close to this minimum level to keep your hands from straining. As said a lower action will also contribute to a lower amount of pressure necessary, and thus even with barbed wire roundwounds your hands will take less friction "damage"