It's not really a specific “English” technique, but there are several ways to describe this:
- To Bowdlerize (Cambridge, Oxford)
Mid 19th century: from the name of Dr Thomas Bowdler (1754–1825), who published an expurgated edition of Shakespeare in 1818, + -ize. [Oxford]
- Political Correctness (Cambridge, Oxford) — not so much the technique as the motive
A politically correct word or expression is used instead of another one to avoid being offensive [Cambridge]
- To Understate (Cambridge)
to describe something in a way that makes it seem less important, serious, bad, etc. than it really is
- Euphemism (Wikipedia — sorry!)
A euphemism /ˈjuːfəˌmɪzəm/ is a generally innocuous word or expression used in place of one that may be found offensive or suggest something unpleasant. Some euphemisms are intended to amuse; while others use bland, inoffensive terms for things the user wishes to downplay. Euphemisms are used to refer to taboo topics (such as disability, sex, excretion, and death) in a polite way, or to mask profanity.
You can find a number of others by looking up downplay or water down and chasing around a thesaurus. One usually found in the negative is [Don't] mince [one's] words.