Jtdylktuy

"Je ne sais pas nager" means "I don't know how to swim," if I'm not mistaken. So how do you say "I know not to swim"? (As when reassuring someone that you are not going to do something, e.g. to a parent "Yes Mom I know not to talk to strangers").

There could be different ways to express that, depending on the context:

Je sais qu'il ne faut pas nager

Je sais que je ne peux pas nager

Your statement, "I know not to swim" is perhaps hard to translate because it is not entirely clear, in English, just what it means, which I think you have implicitly acknowledged by explaining the sort of context where it might be said. I think the slightly different statements "I know I should not swim" or "I know I can't swim" would not require this sort of explanation, and would thus be easier to translate. And their translations, respectively, would then be:
"Je sais qu'il ne faut pas nager" (or "Je sais que je ne devrais pas nager", or "Je sais que je ne peux pas nager") for the first, and
"Je sais que je ne peux pas nager" (or "Je sais que je ne sais pas nager" or "Je sais que je ne suis pas capable de nager") for the second.

Notice that one of these, "Je sais que je ne peux pas nager", retains the ambiguity of your "I know not to swim" - in both the French and the English here, it is not clear whether you will refrain from swimming because it is not allowed or because you are not capable.

"Je ne sais pas nager" means "I don't know how to swim," if I'm not mistaken.

Right.

So how do you say "I know not to swim"? (As when reassuring someone that you are not going to do something, e.g. to a parent "Yes Mom I know not to talk to strangers").

I would use future simple for this:
Je ne nagerai pas (I will not swim)
Je ne boirai pas (I will not drink)

Or similar to Greg's answer, one could also say
Je sais que je ne dois/devrais pas nager (I know that I must/should not swim)

Indeed,

Je ne sais pas nager

translates to "I don't know how to swim". "I know not to swim" would be

Je sais que je ne dois pas nager

But the sentence is weird, even in english lacking a context.
In you strangers example it is clearer :

Oui maman, je sais que je ne dois pas parler aux étrangers

I think the real problem here is that the questioner is stuck on a particular English syntactical construction and trying to replicate it in French, rather than do the obvious, namely, swim about first in English to find other ways of saying it in English. Je ne dois pas nager/je sais qu'il ne faut pas nager/il faut que je ne nage pas sound like they are all well within the questioner's ability, but he/she is just too stuck on one English construction to think of them.