Yes, it is possible if the father transmitted the hepatitis C virus to the mother and the baby contracted it from her (perhaps via placental hemorrhage with blood mixing or during delivery).
Hepatitis C is known to be spread in the following ways: contact with contaminated blood (e.g., accidental blood exposure in a clinical setting; transfusion or organ transplant, especially before 1992); contaminated needles (e.g., IV drug use, tattooing, body piercing; some concern with razors); mother-to-infant transmission during childbirth; sexual intercourse (this is rare though).
Research is ongoing about possible hepatitis C transmission through saliva (e.g., kissing or sharing a toothbrush). Such transmission is thought unlikely, but possible if the person with the virus has a viral load over one million and both persons involved have gum disease.
Data references about hepatitis C:
I reviewed a study regarding detection of hepatitis C virus RNA (viral reproductive material) in raw (untreated) semen and in the centrifuge-separated seminal fluid as well as the sperm cell pellet. It may be possible that the sperm cells themselves can transmit the hepatitis C virus directly to the ovum (female egg cell) and hence the fetus.
Assisted reproduction and HCV reference: http://humrep.oxfordjournals.org/cgi/content/abstract/15/4/810