Fret buzz can come from several places. Most of the time it is the player causing the fret buzz, not the bass. Have you ever noticed when things go wrong you’re always there? Yes, it’s probably your technique. Make sure you are playing right behind the frets with each finger. Don’t play centered between the frets. Secondly, don’t pluck really hard. Basses need an amp for a reason. Turn up the amp and pluck more lightly. You may need a more powerful amp if you find yourself plucking hard to compete with a drummer or (even more often) a guitarist. Check out my other website where you can find my bass technique lessons.
It is natural to hear a bit of buzz coming from the fretboard as you play. The main question is, does it come through the amp? You may hear all sorts of noises that don’t project to the amp. I wouldn’t worry about those.
OK. So you’ve applied the right technique and your bass is still buzzing. It may need to be setup correctly. You can read how to setup a bass. The buzzing comes from the strings rattling against the frets. That means the strings are too close to the fretboard. This is either from the neck not having enough relief (a slight bow to it), the string height at the bridge and/or nut is incorrect or some combination of those problems. This may be due to recently putting on new strings of a different guage. Weather changes (humidity) also affects the straightness of the neck. I suggest you have a professional set up your bass. If you don’t do it correctly, your bass may play seriously out-of-tune and that’s far worse than buzzing!
If you only have one string that buzzes, you may have a low fret somewhere on that string. It may be because you’ve worn one fret down too much or the frets were put on poorly in the first place. Play up and down each fret of the string until you hear the buzzing start (or stop). Examine the fret for excessive wear. You may need a refret from a pro.