I have root bits from the previously grown basil in a soil that is perlite vermiculite coco coir and compost. Is there a way to remove them? If not, then how much new potting soil should I mix in? Moreover, how do I ensure that these root bits don't catch insects detrimental for planting a new plant?
The roots bits and insects are not to be worried. What should worry you is the potential of old used soil to harbour disease that may damage or potentially kill the new plant. You could attempt to sterlise the soil, but most gardeners recommend you always start a new crop in pots with fresh, never used soil. Even if you sterlise the soil, you can not get rid of the salts that are left in the soil from the feriliser used on the basil. Basil are fairly heavy feeders, meaning they will have used up an nutritional value that was in the soil the first time you used.
If you still want to reuse the soil, the roots will just slowly decompose and pose no ill health to the next plant.