Oh, this one is sure to start some wars in some places. Grammatically, we are taught "never to split infinitives" which means "She told me not to smoke." is more correct. It's also the more common way of saying that type of sentence. "X told me not to V." However, there's a lot of disagreement among English experts about whether or not splitting infinitives is wrong. From my perspective, we split infinitives all the time, so I would say that both are correct. However, we sometimes prefer one version (split or not) over others. In short, don't worry too much about it unless you have a job where their chosen style guide forbids splitting infinitives.

'She told me not to smoke' is the correct form. The other form would be marked as wrong in an exam.

Use, "She told me not to smoke." As much as I agree with Johnny's point on splitting infinitives - which is a very old argument anyway - I don't believe that the issue of splitting infinitives is the issue here. The second sentence still sounds "off". You could understand it, but it still doesn't sound right or natural.

I have been learning English in China for over ten years.In our tests "She told me not to smoke" is grammatically correct,while "She told me to not smoke"is definitely wrong.