我个人觉得用"a gentle reminder"比较好点.温馨提示 "a gentle reminder"随手关门 的翻译有很多: Close the door after you, please. 请随手关门。

Shut the door conveniently. 随手关门。

Shut the door after you. 随手关门。

Please close the door as you go out. 请你随手带上门。

我个人认为,"温馨提示 随手关门 "的翻译是:[A gentle reminder] Close the door after you, please.。