Steinbeck seems to be using handy in the conversation as a truncated form of the idiomatic phrase "handy with his fists," meaning in a narrow sense "a good pugilist [in a formal boxing match]" and in a broader sense "a good fighter [in any physical altercation]." This is certainly the conclusion that a reviewer of Of Mice and Men, writing in 1937, draws in his brief description of Curley. From Fred T. Marsh, "John Steinbeck's Tale of Drifting Men," in the New York Times Saturday Review of Books and Art (February 28, 1937), reprinted in Michael Meyer, ed., The Essential Criticism of John Steinbeck's of Mice and Men (2009):
There are trouble makers in the bunkhouse Curley, the boss's son, a little fellow handy with his fists, likes to take on big clumsy fellows, pick fights with them. He wins no matter how the fight comes out, because if he licks the big fellow, every one says how game he is, and if he gets the worst of it every one turns on the big fellow for not taking [on] some one his own size.
Although "handy with [one's] fists" doesn't appear in Christine Ammer's American Heritage Dictionary of Idioms, it is a widely used and recognized set phrase in the United States and perhaps in Great Britain as well (where it probably originated). Here is the Ngram chart for "handy with his fists" for the period 1850–2005:
As you can see, the expression was most popular during the half-century from about 1910 to about 1960, but it has by no means disappeared in more-recent decades.
A Google Books search finds instances going back to 1833 (in England) in the narrower sense and to 1858 (in a U.S. publication, but by an author who spent considerable time in both the U.S. and England) in the broader sense.
From "A Trip to Paris with Mr. Jorrocks," in The New Sporting Magazine (June 1833):
Jorrocks too, as many of the readers of the N. S. M. are aware, is very handy with his fists, having studied under Bill Richmond, the man of colour, and is reported to have exhibited (incog.) with a pugilist of some pretensions, at the Fives-court; and considering all things, fists seemed as proper a mode of settling the business as any we could hit upon.
From Nimrod's Hunting Tours, Interspersed with Characteristic Anecdotes, Sayings, and Doings of Sporting Men (1835):
Our party at the Club, in consequence of the weather, was small, but everything extremely comfortable and correct; and in the secretary (Mr. Benjamin Ord, well known in the school as Ben Ord, and very handy with his fists) I recognised an old Rugbean, although he quitted half a year after I entered the school.
From "January Searle" [George Searle Phillips], Myra, the Gipsy Prophetess, serialized in Frank Leslie's New Family Magazine (May 1858):
"Ye may say that, ye may well say that, ye ill-favored tyke! There's no lad on all the head fit to stand in his shoon. He war as strite as a young poplin tree, and as handy with his fists as the best tawny ye've got in the 'campment; and no chap in these parts could fling him in the wrastling ring, or beat him at the foot race, or wark a cobel, or shoot, or fish, wi' the likes o' him. Wae's me! he's left his old mother many a long year, and gone across to Ingies, and I shall never see him no more."
And from Jack Hyfligher, M.D., and What He Did, serialized in Yankee Notions (April 1859):
They trampled up and down the hall, struggling and swearing, knocking now against the bannisters and now against the wall, until both got exceedingly exhausted. If Jack could have made the unknown stand off, he could have settled the matter very speedily, for as we have seen, he was very handy with his fists.