"Formal, behavior that is accepted as socially or morally correct and proper. The state or quality of being correct and proper" (OED).

For a young, unmarried woman such as Marianne to correspond with an unmarried man was considered extremely improper. In fact, for a single woman to initiate such correspondence would be considered scandalous, thereby tarnishing her virtue and reputation as an eligible bridal candidate. Only if they were engaged to each other would this practice be looked upon favorably; in fact, it might be necessary and expected. If they were not engaged, and especially not to each other, open correspondence would indicate a lack of morals in the woman, in that society would assume the young woman to be the man's mistress. Brandon follows this statement by extrapolating that Marianne and Willoughby would soon be married, since that would be the only alternative so her social reputation would not suffer.