Playing the Total War series, I've found that the one sure counter to horse archers is large numbers of missile infantry protected by spearmen to counter a charge. Did the opponents of the Huns, Mongols and Turks (to name a few) attempt to counter the horse archer's supremacy in the field with infantry archers, and how successful were they in doing so?
The short answer is that they never faced that kind of force.
The longer answer is that large units of unmounted archers were very rare. The English had them but never fought against horse archers. Crossbow units were moderately common but weren't mobile enough or fast shooting enough to be decisive.
There is only one exception I'm aware of. In the battle of Ain Jalut, where the Mamluks finally stopped Mongol advances it is said that archery fire from hills above the Mongol was an important factor in reducing the Mongol force before the final Mamluks counterattack.
More generally, those (few) who succeeded against horse archers used tactics to limit their mobility, either using terrain (Ain Jalut, Kulikovo) or by using their own forces to encircle the enemy (Alexander the Great at Jaxartes).