A compound dissolves in aqueous NaOH only if it is more acidic than water, i.e. its conjugate base is more stable than $\mathrm{O}{\mathrm{H}}^{-}$. This includes carboxylic acids, most phenols (including naphthols), and acidic NH groups; it excludes aldehydes, amines, alcohols, and severely sterically hindered phenols (whose phenoxide cannot be solvated).

From the nine compounds, the ones meeting this criterion are: II (1-naphthol), III (substituted aminophenol with phenolic OH), IV (para-aminophenol with phenolic OH), V (benzoic acid), and VII (naphthoic acid). The benzaldehyde (I), aliphatic amine (VI), the di-tert-butyl hindered phenol (VIII), and hindered naphthol (IX) do not dissolve.

Total $=5$.