The expression “off one’s base” has a literal meaning in baseball, referring to a position taken by a runner away from the bag or base he has occupied, but in the slang sense it has nothing to do with baseball.
Here base pertains to that which supports a person, and the expression thus means mentally unbalanced; crazy; off one’s rocker or bean; screwy.
One of the earliest to use the expression in print was George W. Peck, author of Sunshine (1882) and Peck’s Bad Boy and His Pa (1883).
In Chapter II of the latter:
“… the boy has been for the three weeks trying to think of some innocent joke to play on his father. The old man is getting a little near sighted, and his teeth are not as good as they used to be, but the old man will not admit it, and he would bet a hundred dollars that he could see as far as ever. The boy knew the failing, and made up his mind to demonstrate to the old man that he was rapidly getting off his base.”
The means taken, incidentally, was that the bad boy cut up some small rubber hose and, with the connivance of the “hired girl,” mixed it into his pa’s serving of macaroni.
The mischief was eventually discovered, and the boy had another session in the woodshed.
