Hint: Absence of $d$-orbitals in boron. $\mathrm{BCl}_3$ is electron deficient species and has $6 \mathrm{e}^{-}$in the valence shell of central atom B. But it does not form dimer to complete its octet due to small size of boron. Bridging $\mathrm{Cl}$ comes very closer in dimer and dimer becomes unstable. Therefore $\mathrm{BCl}_3$ exists as monomer.
$\mathrm{AlCl}_3$ has the presence of 6 electrons, so, to complete its octet it exists as dimer. It has vacant $3 d$ orbitals, so it can extend its coordination beyond 3. Hence, it completes its octet by forming coordinate bond with $\mathrm{Cl}$ atom of the other $\mathrm{AlCl}_3$


Structure of anhydrous $\mathrm{Al}_2 \mathrm{Cl}_6$