Step 1: Understand coordinated water.
Coordinated water refers to water molecules that are directly bonded to the central metal ion in a coordination complex. These water molecules are part of the coordination sphere.
Step 2: Analyze each compound.
\begin{itemize}
\item I. CrCl$_3$.6H$_2$O: This compound exists as a hydrate where some or all of the water molecules are coordinated to the chromium(III) ion. The formula can be written as [Cr(H$_2$O)$_6$]Cl$_3$ or isomers with some Cl$^-$ ions inside the coordination sphere. In either case, some water molecules are coordinated.
\item II. BaCl$_2$.2H$_2$O: Barium is an s-block element and typically forms ionic hydrates where water molecules are held by electrostatic forces in the crystal lattice but are not directly coordinated to the metal ion in a complex.
\item III. CuSO$_4$.5H$_2$O: Copper(II) sulfate pentahydrate has the structure [Cu(H$_2$O)$_4$]SO$_4$.H$_2$O. Four water molecules are coordinated to the copper(II) ion, forming a complex ion, while one water molecule is a lattice water. Thus, it has coordinated water.
\end{itemize}
Step 3: Identify the compounds with coordinated water.
Compounds I and III have coordinated water molecules.
Final Answer:
\[
\boxed{\text{I, III only}}
\]