IMO it is more about where you go and less about what the degree is. In general, the more linguistics you know, the better you will be positioned to do this kind of work. A basic undergraduate degree where you learn the essentials is all that a lot of field workers have, but unless you're associated with SIL and get special training, it is unlikely that you will be competent to take on an arbitrary language. With a PhD, what you will hopefully get is training in "how to figure it out", but not "the answer".
On the face of it, it would seem that you ought to focus on phonetics and phonology, because you need to develop skills at reducing speech to phonetic symbols (if the language has [k, q, k', q'] and they all sound the same to you, that is a problem). Abstractly learning technical terminology is not very useful, what is helpful is taking a field methods class of two, on Tigrinya and Buginese for example. However, a bit of reality check on practical orthography is also necessary. Lushootseed spelling is a stunning counterexample to the generalization that people don't want strange letters in their language (it helps that the promulgator of the writing system was a speaker). Many languages devise orthographic work-arounds to avoid phonetic symbols. So some training in realities of practical spelling systems is good.
Often, SIL people take their undergraduate degree plus some internal training, then go to the field and work for some while, take a break and get an MA, more work, and sometimes a PhD. The important point is that you don't know how to gather and analyze useful data without some training on that area, and the typical intro to syntax and phonology class will not tell you how to do that either. Indeed, a PhD from certain universities would be completely useless; and a BA from other universities would be quite useful. If the institution has a regular field methods class, you are look at the right type of institution. If it is a year long class, you have landed in heaven.