http://www.goodreads.com/quotes/171666-the-central-bank-is-an-institution-of-the-most-deadly
“The central bank is an institution of the most deadly hostility existing against the Principles and form of our Constitution. I am an Enemy to all banks discounting bills or notes for anything but Coin. If the American People allow private banks to control the issuance of their currency, first by inflation and then by deflation, the banks and corporations that will grow up around them will deprive the People of all their Property until their Children will wake up homeless on the continent their Fathers conquered. ”
The quotation itself is a fake. The official Monticello website collects spurious Jefferson quotes and is a pretty authoritative source. You'll find their entire reasoning here. While elements of the quotations are based on Jefferson's words, it appears that the quote was invented in the 1930s. Interestingly, you have a somewhat different version than those provided by the website. Note that in the quote TJ predicts a dire future: this is a common tactic among fakers.
So the question is how did Americans in the early Republic feel about banks. Maybe the best place to go would be the debates over the chartering of the First Bank of the United States. Treasury Secretary Alexander Hamilton recommended a central bank in his report of December 1791. It would be a source of funds and credit, and its banknotes would become a national currency. This bank proposal had solid support in both the House and the Senate, among both Northerners and Southerners. Some Congressmen were opposed to the bill, for various reasons. Some worried that it would put so much wealth into private hands that they could bribe the politicians. Others said the bank favored mercantile interests too heavily. For some (and Madison and Jefferson probably fall into this camp), this rhetoric was a cover for a sectional fear: the bank HQ would be in Philadelphia, but the new national capital was to be located in the South. Some southerners were worried that a Northern bank would retard or prevent the establishment of a capital on the Potomac. So we can't necessarily take the politicians' words at face value. For his part, Jefferson argued that the BUS violated the Tenth Amendment. Hamilton retorted with the concept of implied powers.
The bill originated in the Senate and passed in the House by a vote of 39-20.
Of course, ideas about banks and banking would evolve over time. A generation later, a more democratic America, having lived with a central bank for some time, chose Andrew Jackson to destroy this perfidious monstrosity.
Source: Elkins and McKitrick, The Age of Federalism. If you're interested in a wider understanding of this topic, Bray Hammond's Banks and Politics in America is a classic.