He made the correct wager.
He was in second place going into Final Jeopardy, so the only way he could win was if the woman in first got Final Jeopardy wrong (assuming she made a large enough wager to cover him if they both got it correct -- which he should assume she did).
As a result, the proper bet for him is one that ensures he finishes ahead of the third place player if both he and the first place player get Final Jeopardy wrong -- which is exactly what he bet.
Going into FJ:
- Emma = $26,600
- James = $23,400
- Jay = $11,000
Emma's proper bet is $20,201 to ensure she wins if she gets it right (that is what she bet and she guaranteed her victory by getting FJ correct). She finished with $46,801. If James had bet everything, he would have lost by a dollar.
If Jay had bet everything, his maximum possible score was $22,000.
Because James had more than $22,000, his proper bet was $1,399, to ensure he'd finish with $22,001 if he was wrong. That is what he bet.
The only chance James had to win was for Emma to get FJ wrong. If she had, she would have finished with $6,399 and James would have won even if he got it wrong too.