That's the idea, but if I remember correctly there's not a whole lot of evidence suggesting that more borrowing happens when the rate is low, for the reason mentioned above: you don't borrow money for fun when it seems cheap, you borrow money because you need it. Borrowing isn't particularly price sensitive.
(It is in the margin in the sense that more doubtful gambles pay off when borrowing is cheaper, but the large part of borrowing happens because of an immediate liquidity demand, not a desire for leverage.)
I think that's what they call "pushing on a string" (1).
Central banks can make lending/borrowing a worst business (higher interest rate) but they can't force people to borrow.
So, monetary policy is very limited when demand falls, then fiscal policy is necessary, but, until recently, fiscal policy was anathema because of ideological reasons.
(It is in the margin in the sense that more doubtful gambles pay off when borrowing is cheaper, but the large part of borrowing happens because of an immediate liquidity demand, not a desire for leverage.)