Ransom!
(aka Fearful Decision)

1956

After David Stannard (Glenn Ford) goes on live television and announces that he will not pay the ransom, and then offers the $500,000 intended ransom amount as a bounty on those involved if any harm comes to his son Andy (Bobby Clark), the kidnappers (never seen) apparently get cold feet and release Andy. The bloody shirt that the police found got that way when Andy bit the "nurse" who helped nab him.

This movie was remade in 1996.
Jodun306 has contacted us with the information that this is Episode 41 of the US Steel Hour, from May of 1955.

Thanks, Jo!