In my experience, DTT is added simply to keep cysteine residues from oxidizing. Since ligase is a cytoplasmic protein, it is normally in a reducing environment. In the reaction, this reducing environment is maintained by DTT.
I think DTT acts just as a reducing/antioxidant agent - it is thiol-specific, but can react in common Red-Ox reactions as well. It can maintain a correct enzyme conformation, e.g. by preventing Cys oxidation or cross-linking with S-S bond formation resulting in the decrease or loss of activity. I don;t really see how DTT could participate in the ligation reaction itself (phosphate transfer etc.) Perhaps it is also true for many other enzymes of NA biosynthesis (polymerases, topoisomerases etc.) where DTT is also present in reaction buffers. And not only for NA enzymes.
In my experience, DTT is added simply to keep cysteine residues from oxidizing. Since ligase is a cytoplasmic protein, it is normally in a reducing environment. In the reaction, this reducing environment is maintained by DTT.