How to calculate the composition of a borate buffer with a defined pH using the Henderson-Hasselbalch equation?
Boric acid and sodium borate forms an acidic buffer (A mixture of weak acid and salt of weak acid and strong base). The acid is weak hence the concentration of basic ion from acid is much less compared to the concentration of basic ion obtained from the salt which can ionize completely. Yes. Some of the acid dissociates to form hydronium ion and a basic ion. That's how a buffer works.

Consider a weak acid HA and the salt BA.
Its easy to show that the concentration of $H^+$ due to ionisation of HA is $$[H^+]=k_a\frac{[HA]}{[A^-]}$$
It is reasonable to assume that the concentration of $A^-$ from BA is much larger than that from HA due to complete ionisation of BA. Hence $[A^-]=[BA]$=[salt]
So, $$[H^+]=k_a\frac{[acid]}{[salt]}$$
Taking logarithm, you get the desired equation.