It seems that "doubt whether" was far more common for a long time, but the usages have largely converged, and both appear to have currency.
Personally, I prefer "whether" in both of these cases, but I would not see a big distinction between them. Neither usage would mark a speaker as being non-native or particularly regional to me, nor seem out of place. I would find "if" in these sorts of sentences to be somewhat more casual, however.
I am not aware of any rule governing which one to use... if you're in doubt, I would go with "whether" when you are discussing a true-or-false/yes-or-no condition in an object clause like this.
You didn't ask, but note that "I doubt they'll make it" is also acceptable. That's particular to the verb doubt though.