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In Tokyo, I saw this sign next to the smoking area.

It says

Please smoke with good manners in the area surrounded by planters.

The part "with good manners" sounds a bit weird to me. Is it just me, or is it not idiomatic?

One can have/teach/forget manners

He dressed well and had impeccable manners.

They taught him his manners.

I'm sorry, I was forgetting my manners.

but "with manners" seems a bit off, doesn't it? How would you say it?

While you can do things with good manners, it is rare to direct someone to do so in imperatives. Unless addressing children, it is not required to remind people to be polite— a person with good upbringing has good manners out of habit; an poorly raised person cannot tell the difference, anyway.

This may reflect a difference of cultural perception, but I think more likely that it is simply a poor translation from Japanese. A search on "smoke with good manners" mostly returns results from Japan. Considerate would be the more idiomatic adjective to use to remind people to think of others, so a similar campaign in an English-speaking city might be please be a considerate smoker or please be considerate when smoking, phrasings which are common in such signs.

When speaking generally of politeness, consideration, and other positive social behavior, good manners are something you have, not something you do (as with one's habits or one's nature). Someone with good manners may be said to be well-mannered, and someone without good manners is ill-mannered; it is more usually expressed that someone has good manners, than that they do something with good manners.

It is not uncommon to see "with good manners". For example see Google and Google Books. You can learn good manners, practice good manners, or keep good manners.

A Japanese speaker might offer a better phrased translation, but this doesn’t seem too bad.

One can use good manners in other actions. A smoker can be mindful of those nearby and carefully dispose of the remains. That would seem to be good manners. As others noted, “be considerate” would be how one would word this in English, typically.