Not sure but I think the stringy looking things are future flowers. IME basil turns bitter when it bolts. I cut off the top of plant to refresh

That is dodder, a parasitic plant.....looks a lot like silly string. You need to remove it ASAP.

Basil flowers look quite different, even when forming. As a member of the Lamiaceae, the flowers will look very similar to those that appear on coleus. Or mint.

How is the best way to remove it? I pull them off every other day but they come back.

I stand corrected gardengal. I had never heard of dodder until this thread but I did seem to recall flowers were "longish" thus my erroneous guess.

rather than to continue the battle, would it make sense to get a new plant and soil and start over?

I think that's what I'll do. I don't want it to spread to my other plants.

bossy, basil flowers look just like those of any mint.

efreni, I will suggest that you clear that entire area of everything in order to have any hope of getting rid of the dodder. How large an area has it infested and how long has it been a problem? What kinds of other plants do you see it on? Finally, what's your general location?

If it's at the top of the basil, I would just cut it way back, using the cuttings for pesto or to freeze or dry, of course removing any infected parts. Then see if it returns in the new growth. There is no advantage to leaving the basil growing. If it bolts too much it reduces the quality of all the leaves on the plants.