What does Soapy fear?
The quick answer is that Soapy fears the onset of winter, considering that he is homeless and lives on the street.
On the previous night three Sabbath newspapers, distributed beneath his coat, about his ankles and over his lap, had failed to repulse the cold as he slept on his bench near the spurting fountain in the ancient square.
However, O. Henry makes it clear that he fears more than this. Soapy knows that there are various services available to the homeless in winter.
There was an endless round of institutions . . . on which he might set out and receive lodging and food accordant with the simple life.
However, Soapy feels that accepting these services is humiliating.
If not in coin you must pay in humiliation of spirit for every benefit received at the hands of philanthropy.
So, Soapy fears being forced by the cold to accept charity. His preferred plan? He wanted to be sent to "the Island", a prison named Blackwell, where he had spent the last 3 winters. As Soapy sees it,
[I]t is better to be a guest of the law, which though conducted by rules, does not meddle unduly with a gentleman's private affairs.