Fabric finishing involves applying various agents to improve a cloth's look, feel or performance, and stiffening is one such finish meant to add firmness and body to a fabric that would otherwise hang too limply.
For silk specifically, this stiffening is achieved using gelatin. During finishing, a dilute gelatin solution is brushed or padded onto the fabric and then dried, leaving behind a fine protein coating between the fibres.
Because silk is naturally a very fine and soft fibre, it needs a gentle sizing agent rather than a harsh chemical one, and gelatin fits that requirement well, adding crispness while still allowing the fabric to retain its characteristic drape and lustre. This makes it the standard choice among stiffening agents for silk in fabric finishing.
Therefore, the correct stiffening agent used for finishing silk fabrics is gelatin.