I have a set of variables coded as binomial.
   Pre VALUE_1 VALUE_2 VALUE_3 VALUE_4 VALUE_5 VALUE_6 VALUE_7 VALUE_8 
1   1       0       0       0       0            
        A quick solution would be something like
Res <- cbind(df[1], VALUE = factor(max.col(df[-1]), ordered = TRUE))
Res
#   Pre VALUE
# 1   1     6
# 2   1     5
# 3   1     5
# 4   1     5
str(Res)
# 'data.frame':  4 obs. of  2 variables:
# $ Pre  : int  1 1 1 1
# $ VALUE: Ord.factor w/ 2 levels "5"<"6": 2 1 1 1
OR if you want the actual names of the columns (as Pointed by @BondedDust), you can use the same methodology to extract them
factor(names(df)[1 + max.col(df[-1])], ordered = TRUE)
# [1] VALUE_6 VALUE_5 VALUE_5 VALUE_5
# Levels: VALUE_5 < VALUE_6
OR you can use your own which strategy in the following way (btw, which is vectorized so no need in using apply with a margin of 1 on it)
cbind(df[1], VALUE = factor(which(df[-1] == 1, arr.ind = TRUE)[, 2], ordered = TRUE))
OR you can do matrix multiplication (contributed by @akrun)
cbind(df[1], VALUE = factor(as.matrix(df[-1]) %*% seq_along(df[-1]), ordered = TRUE))