When I write bass lines, I find that they sound better with approach notes but I also find that sometimes the approach notes clash with or clutter certain run-in vocal phrases that start with a few quick syllables before the start of the bar. Has anyone else had a similar problem? What is your take on this?
Yes I've definitely had this problem. There are probably quite a few ways to deal with it. For starters you can simplify the bass line, or if you want to keep it then simplify the vocal instead. Or you can keep your bass line and alter the vocal so that it fits in with the timing and tonality of the bass.
Try just one approach note. Make sure it fits with what the vocals are doing (i.e. don't play a tritone or a semitone off of them). This idea is sometimes called a "burp" in jazz basslines.
If you can figure out exactly what the vocals are doing, sometimes it sounds good to double or harmonize with them.
Otherwise, try writing with ending phrases rather than approach phrases. At the end of the day, the bass's primary roll is to be the foundation for the band. We all want to stick out like Flea or Les Claypool, but that isn't always what the song needs. I try to put the song before my ego, and I find that my bass lines usually fall into place quickly after.
One of the things about playing bass is that one has to be aware of all of the other parts of the music that one interacts with (okay, unless you're a solo bass player). As a bass player, one has to define harmony with the bass note, and also to hold a strong rhythm down. The question becomes one of balance: Can I add these notes and still do my job? By adding complexity, do I still have enough responsibility to make sure it fits well with the other parts?
Often times, the approach notes won't fit and it'll clash. How to fix this? Well, certaintly making simpler music will work, but I get the sense that that's not what OP is going for. One has to be able to fill the space in the arrangement. For example, if there's a nice empty section between lines in the verse, maybe a nice approach will work out. If the singer or other instruments are also taking these empty spaces to add in fills of their own, backing off might be a better option. One thing that often impresses me is when performers are so locked-in that two or more players will do the same fill spontaneously at the same time live! That is a sign of amazing coordination. With a lot of experience, one can learn to know when to interject these approach notes, and often more importantly, when not to, just by listening and getting to know one's fellow performers well.
It's not just a question of whether you can execute the phrase, it's a question also of whether the phrase serves the music. One learns the ability to judge these aspects through musical experience, and a good knowledge of theory of course will help.