The actual quote which is attributed to Mayer Amschel Rothschild is:
Permit me to issue and control the money of a nation, and I care not who makes its laws!
A number of sources (such as this one) claim that this statement was made in 1838 (which would have been a difficult feat as he would have been dead for 26 years by then). Wikiquote claims that there is no way to verify by whom, when or why it was made. It notes:
No primary source for this is known and the earliest attribution to him known is 1935 (Money Creators, Gertrude M. Coogan). Before that, "Let us control the money of a nation, and we care not who makes its laws" was said to be a "maxim" of the House of Rothschilds, or, even more vaguely, of the "money lenders of the Old World".
It is adapted from another well known quote:
Let me make the songs of a nation, and I care not who makes its laws.
This is in turn attributed to the Scot, Andrew Fletcher:
In An Account of a Conversation he made his well-known remark "I knew a very wise man so much of Sir Christopher's sentiment, that he believed if a man were permitted to make all the ballads, he need not care who should make the laws of a nation."