Which verb should I use and why?
Either the wild dogs or the baboon makes or make that howling sound.
I think since dogs is plural it should be make?
There's a tricky rule when the conjunction in a compound subject is "or" --
The verb needs to match the subject that is closer to it in the sentence; in this case, "baboon" is closer to the verb, so you use the singular verb form.
See #5 here:
http://grammar.ccc.commnet.edu/grammar/sv_agr.htm
Yes, you are correct. In this sentence, the subject is "either the wild dogs or the baboon" which is a compound subject joined by the coordinating conjunction "or". When you have a compound subject joined by "or", the verb should agree with the closest subject.
In this case, "the wild dogs" is the closest subject to the verb. Since "dogs" is plural, the correct verb to use is "make", which agrees with the plural subject. So the correct sentence would be:
"Either the wild dogs or the baboon make that howling sound."