I'm working on a Halloween costume of taffeta, and it's my first time working with this fabric. Because it's just a costume, it's an inexpensive polyester taffeta, but obviously I still want it to look good. I just started sewing the actual taffeta pieces today, and all my seams make little striations, like thread runs, in the fabric. I read on one page online that a ballpoint needle may solve this problem, but the ballpoint seams look the same as the fine, sharp needle seams. They are most noticable in the seam allowance, and maybe with some gentle rubbing and pressing, they won't be too noticable in the finished piece. But does anyone have any other suggestions or tips that I can try?

No tips--that's boogery stuff to work with. Since it's for a costume, could you glue or fuse the seams? Good luck.

All I can suggest is using the tiniest needle your thread will allow - like a size 10 or 12. I'd stick with sharp needles because ball point is usually for stretch fabrics like knit. If you have a problem with fraying - FrayCheck is marvelous. It works like clear glue and stops the threads from fraying.
Also, since it's "only" a costume, you could try using a longer stitch so you don't have as many needle holes.
Sometimes just ironing it afterwards helps a lot too.

It sounds to me like you have a little snag on your needle. I'd try a brand new needle.

It is for things like this that I love my serger! Hope you find a way to make the needle work. So frustrating.

Ditto to others ..... new - sharp - fine - needle (not ball point) and a little longer stitch than normal

Yup, a serger is great with this kind of material and hassle.

I read before I started the project about the fraying issue, and was prepared for that, and this taffeta actually hasn't been too bad--I've worked with much worse. It's really the runs in the fabric from the needle that bother me, even though I'm using exactly the needle recommended for it. A woman at the fabric store told me that a really sharp needle makes all the difference, so I tried sharpening my new needles with the little strawberry pin sharpener on my pin cushion--and it worked really well! I can still see some of the runs, but they're not nearly so noticeable, and after a pressing, they don't show up at all. I'll have to post some pics when I have the project done. Thanks for all your suggestions!