"I don't drink"
That would literally mean "I am not (the action of) drinking".
Remember that the ku- form of the verb is the infinitive/gerund, which is essentially the name of the action. For example, you can say "Kula si kunywa," meaning "Eating is not drinking" - talking about two different actions here, or "Kunywa ni muhimu," meaning "Drinking is important," etc.
When we say "I am not drinking", we're not saying that you're not the action of drinking. It's the way we form the present continuous tense. The -ing form of the verb here is the present participle, which just happens to be identical to the gerund in English (both end in "-ing"), and I think that's why you're getting confused.
So to make this sentence, you have to use the negative present tense in Swahili, which is formed with three things:
(1) The negative subject prefix: si-, hu-, ha-, hatu-, ham-, hawa-, hau-, hai-, hali- etc.
(2) The verb stem (WITHOUT the ku- extention for monosyllabic verbs), so in this case, it's -nywa
(3) Verbs that end with a final "a" change this to -i to indicate "This is present and negative." Verbs that end with another vowel (only words borrowed from other languages, mostly Arabic) don't do anything here.
So that gives us si-nyw-i.
There are two syllables. Si-nywi
The stressed syllable is SI (like English SIT, https://en.wikipedia.org/wiki/Voiceless_alveolar_fricative#Voiceless_alveolar_sibilant), then the second syllable is NYWI (this is a combination of the consonantal sound of NY+W and the vowel I, where NY is palatal nasal like Spanish Ñ / https://en.wikipedia.org/wiki/Palatal_nasal)
Please check the syllables and word sounds on this: https://www.youtube.com/watch?v=IUjpXWPaWE0 (Kiswahili Sounds - Learn how to speak Swahili)
In this video, the sound combined with the vowels (NYWA/NYWE/NYWI) and the verb KUNYWA are shown in 10:24