According to the online Merriam-Webster, Mocha is a small town and seaport in southwest Yemen that as early as 1773 had already given its name to the style of coffee.
The Arabic pronunciation of the name of the town can be heard here. However, nobody in the English-speaking world says the name that way, and I therefore doubt that your local Starbucks barista would understand what you meant if you used the Arabic pronunciation when ordering your coffee.
What must be remembered is that as soon as the speakers of Language A adopt a word from Language B, there is absolutely no guarantee that they will refrain from mutilating the original word in every possible way, from pronouncing it differently, to pluralizing it differently, to using it to mean something different to what it denoted in Language B.
And that is all perfectly natural and OK — it's one of those evolutionary mechanisms that operate in all languages whose speakers come into contact with outsiders.
Anyway, all this is a preamble to my conclusion, which is that there is no 'proper' way in English to say mocha — or bourgeoisie, or Venezia, or gulag, for that matter. Once a word has been imported from Language B by Language A, the speakers of A own it when they are speaking their own language, and they can do what they like with it. And they do. The different pronunciations of mocha around the world that Kyle has linked to in his answer amply demonstrate that point.
All one can say regarding its 'proper' pronunciation by speakers who have imported it into a language other than the one it came from (in this case, Arabic) is that there is likely to be a period where several different pronunciations of the newly-adopted word coexist, and that a consensus may eventually emerge regarding its usual pronunciation among the speakers of the adoptive language. That consensus too may vary from region to region and dialect to dialect.
Indeed, as far as mocha is concerned, I think Janus B J's comment above pretty much covers the situation as it currently applies to most native speakers of British and American English. I have reproduced the relevant part of that comment here, in case it gets deleted:
The standard pronunciation of mocha in English is [ˈməʊkə] in British English and [ˈmoʊkə] in American English. Long /o/ does include a w-ish offglide (the [ʊ]), and a native English speaker would automatically add that in—so if you’re actually pronouncing it something like IPA [ˈmoka] (which is how the Spanish pronounce it, for example), then a native English speaker might find it somewhat difficult to understand.
But I wouldn't advise anyone to go around 'correcting' other people's pronunciation of the word, because they frankly wouldn't have a leg to stand on if they were challenged.