Yes, the magnitude T₁ of the force exerted by the string on block 1 is less than the weight W₂ of block 2.

Explanation

Have a look at the free-body diagrams for the two blocks in the diagram above.

One important thing to know is that the tension T in the string is the same on both sides of the pulley, since the pulley is massless and frictionless. This tension is equal to the magnitudes T₁ and T₂ of the forces exerted by the string on blocks 1 and 2, respectively. Thus, T = T₁ = T₂.

Now, block 2 is accelerating downward. Therefore, by Newton's second law, the net force acting on block 2 is downward also. Thus, T₂ must be less than the weight W₂ of block 2, as shown in the free-body diagram for block 2, and thus T₁ must be less than W₂ also.