Muting becomes second nature once you've been at it while, but in my limited experience with teaching it's fairly universal that players starting out struggle with it quite a bit.
It's important to understand that it means little to play the correct note if something else is coming out inadvertently. :scowl:
Regarding new students and muting - or just plain left and right hand techniques in general - I've found there are a few common things.
Way too heavy fretting technique. The tendency to grip the neck like a baseball bat and choke the life out of every note. This is usually aggravated by poor setup and high action.
It's a given that 'if it works, it works' but there is something to be said for learning proper left/fretting hand technique which is placing the thumb in the middle of the back of the neck, gently curving the fingers over the strings and keeping them close to the strings. When fretting, you press just hard enough to make the note speak by pressing just behind the fret - not right on top, not in between the frets.
Now you need to have action that allows for you to press the string lightly and get a note - but if you have your bass set up properly, this should not be a problem.
You want to keep all fretting fingers close to the strings and avoid what I see all beginners to where they fret a note (within an inch of it's life) then the finger flies off the string like it was propelled off of a trampoline and the next finger comes down onto the board like a dive-bomber and again frets a note within an inch of it's life.
It shouldn't take a lot of pressure to get a note to speak clearly and you should practice using as light a touch as you can. Volume is your friend. You don't have to attack the strings with all your might to get an aggressive sound.
Right hand technique is equally misguided in many beginners. Because there are so many examples of extraordinary players who really do have very heavy-handed techniques - like Steve Harris and even Geddy Lee - a lot of beginners think that's the best way to attack their instrument. That usually results in 'clanking' (as my teacher called it) - where the strings are forced into the frets by the plucking hand generating metallic 'clanks' with every note. In some instances that's desired - in MOST it's not. If your plucking technique does that 100% of the time, then about 90% of the time you are making way too much extra noise.
A good teacher - and a good setup will go a long way toward rectifying all the extra, unwanted noise that comes from a poor setup and uncontrolled left and right hand technique.