I think these are
the five basic tastes perceived by the tongue: sweet, sour, salt, bitter, and umami.
Five siblings travel to where language is born.
Siblings because they are all the same kind of thing; language is, in some sense, born on the tongue.
The first is kindhearted, and responsible for many deaths.
This must be sweetness. We call people "sweet" when they are kind and pleasant. But in the modern affluent world, where we have ready access to sugary foods, it turns out that they're very bad for us and lead to heart disease and diabetes and other terrible things.
The second wears a frown, yet is hardly negative.
My best guess is that this is sourness. Sour foods make the mouth pucker up, which is kinda frown-like. And sourness is the taste of acidity, which comes (kinda) from hydrogen ions, which are not negative but positive. (I am not certain about this; it might be bitterness instead.)
The third comes to those who are nervous or upset.
That would be saltiness, which is found in sweat (when nervous) and tears (when upset).
Fourth is the unfriendly one. They're sharp, so wake up!
This, then, would be bitterness, which when applied metaphorically generally implies unfriendliness. Bitter tastes are kinda sharp. One bitter-tasting thing that many people consume a lot of is coffee, which wakes you up. (I am not certain about this; it might be sourness instead.)
Last is the rich sibling who cherishes the moment.
This would be umami, the taste-and-feel of glutamate ions, which we get from things like meat and Parmesan cheese. It contributes to the impression of richness in food. To cherish a moment is to savour it, and the umami taste is sometimes described as "savoury".
Surely you have met this family before,
But how, if they have no faces for you to see?
By taste, rather than by sight.
Oh, and the title:
They won't offend you because they are tasteful.
(My thanks to Mike for offering a hint or two in TSL chat.)