Based on suggestions from @jaimedash and @Old_Mortality I found a solution:
DF$Den <- ifelse(DF$Denial1 < 1 & !is.na(DF$Denial1) | DF$Denial2 < 1 &
!is.na(DF$Denial2) | DF$Denial3 < 1 & !is.na(DF$Denial3), "0", "1")
Then to ensure a value of NA if all values of the conditional variables are NA:
DF$Den <- ifelse(is.na(DF$Denial1) & is.na(DF$Denial2) & is.na(DF$Denial3),
NA, DF$Den)