"So, you want Asian salad?"
The Easiest way to correct someone subtly is to repeat what they said, but in the correct way. You're not correcting them, and you get to show them the proper way to say it.
Not every dictionary has phonetic transcript. Also, dialects can vary by quite a large amount. I mean for example, look at American and British English. That means that there isn't really such a thing as 'correct' pronunciation.
There's mispronouncing and then there's ignorance. Surely every adult would be able to pronounce "Asian". I would understand if English isn't their first language.