As suggested, I post here my answers.
This is is with apply. the df[-1] is to exclude the first column (which is not numeric), the x[x == 1] is to subset the elements of x (a single row due to the 1 of the apply) with only values of 1.
 df$TOTAL <- apply(df[-1], 1, function(x) sum(x[x == 1], na.rm = T))
Another (I bet much faster and) easier to code way in base R is:
df$TOTAL <- rowSums(df[-1] == 1, na.rm = T)
both have as a result this
df
  subjectID A B C D E F G  H  I  J TOTAL
1      S001 1 1 1 1 1 0 0 NA NA NA     5
2      S002 1 1 1 0 0 0 0 NA NA NA     3
Data
df <- structure(list(subjectID = structure(1:2, .Label = c("S001", 
"S002"), class = "factor"), A = c(1L, 1L), B = c(1L, 1L), C = c(1L, 
1L), D = c(1L, 0L), E = c(1L, 0L), F = c(0L, 0L), G = c(0L, 0L
), H = c(NA, NA), I = c(NA, NA), J = c(NA, NA)), .Names = c("subjectID", 
"A", "B", "C", "D", "E", "F", "G", "H", "I", "J"), class = "data.frame", row.names = c(NA, 
-2L))