Not quite correct -- and the 'not-quite' is particularly relevant. First off, being president has nothing to do with it (we have non-presidents on plenty of currency and coins, such as Benjamin Franklin and Susan B. Anthony).Sadly - it is in the eyes of the world.
No living president is allowed to have US currency with his image on it. Only past - DEAD - presidents are afforded that privilege. Same is true of Mount Rushmore.
31 U.S. Code § 5114 (b): United States currency has the inscription “In God We Trust” in a place the Secretary decides is appropriate. Only the portrait of a deceased individual may appear on United States currency and securities. The name of the individual shall be inscribed below the portrait.
This prohibition (originally 14 Statute 25) was first passed in 1866 following Spencer Clark managing to have his portrait appear on the 5-cent note. But he wasn't the only person to have appeared on currency while alive, but Clark's appearance was seen by Congress as being blatant self-promotion and abusing his position to do so, prompting them to pass the legislation.
The tricky part is that 31 U.S. Code § 5114 applies to currency and securities, but not to coins, which are treated under a different statute, namely 31 U.S. Code § 5112. The prohibition against living individuals appearing on coins are not blanket, and instead each type of coin has it's own rules. In particular, the prohibition does not apply to commemorative coins and there is precedent, including the 1926 commemorative half dollar depicting Calvin Coolidge, who was not only living, but was the sitting president. Making it an even more relevant precedent is the fact that that coin was issued as part of the Sesquicentennial (150 year) celebration.

