Can hepatitis C be transmitted by a carrier's saliva, or is it only transmitted by blood or sexual contact?

Only by blood. Hepatitis c is usually transmitted by blood, including using an infected person's toothbrush or razor, or getting a tatoo using non-sterile instruments. Trans-nasal trassmission is also possible. Sexual transmission is controversial; possibly limited to rape or violent sexual acts where bleeding is encountered.
Low Risk. A study has shown (oral dis. 2005 jul;11(4):230-5.) that we cannot prove that transmission is impossible. Hepatitis c is believed to be transmitted only by blood. However, unlike many other blood borne viruses (like hiv) virtually any source of blood or blood products seems to be capable of carrying the virus.
See below. While viral particles can be found in saliva, it is not an important mode of transmission of hepatitis c. To my knowledge, transmission through saliva has never been documented. Even sexual transmission is fairly uncommon, but does occur. Among married couples where one partner has hepatitis c, the lifetime risk of the other partner acquiring infection is only 1-2%.