Learn English – Is it ‘thing’ or ‘things’

The verb must agree with the subject of the sentence, not the predicate nominative. In my AmE experience, the word "standards" isn't used as a collective noun. Therefore, the grammatical way to refer to high standards, as a whole, must be "high standards are."

The predicate nominative, which describes the subject, can be either singular or plural. So grammatically, both "a good thing" and "good things" are acceptable. However, there is a small difference between using "good things" or "a good thing":

High standards are a good thing.

This phrasing suggests that the high standards act more as a unit, and that the group (or presence of the group) is a good thing. Like mentioned in mplungjan's answer, an alternative group-based phrasing is "Having high standards is a good thing."

High standards are good things.

This treats each standard a bit more independently, perhaps suggesting that each of the standards should be counted or considered separately. This treats standards more like things that one could list, for example as used on a profile or résumé.

If all of this seems too particular, you could just settle with using a predicate adjective and say "High standards are good."