Green’s Dictionary of Slang

soft money n.

also soft, the soft

1. [mid-19C+] notes, bills, paper money; also attrib.

2. [mid-19C+] (US) currency that is likely to lose its value.

3. [1910s+] (also soft dough) money that is easily earned or otherwise gained.

In phrases

do soft (v.) (also pass the soft, shove the soft)

[mid-19C] to pass counterfeit notes.