(a) Yes, basic solutions also have H+ (aq) ions.
Generally, bases generate hydroxide ions when they are dissolved in water. Basic solutions also have H+ (aq) ions, which are obtained from the ionisation of water. The amount of H+ ions in basic solutions is very less compared with the amount of OH- ions. Hence, they are basic in nature.
(b) When a solution becomes more acidic, the pH becomes lower.