Which Anemone in particular?

I think we need to establish precisely what anemones you're speaking of - if you're in New Zealand, you must be around midsummer, and that's very early for Japanese anemones to not only have flowered, but be finished. Japanese anemones usually means Anemone hupehensis or Anemone hybrida, so you might want to google those and look at images to see if they're the plants you mean.

Otherwise, the next two commonest forms of anemone are actually rhizomes - Anemone blanda and anemone coronaria, usually 'St. Brigid' varieties. If its either of these, you don't need to do anything, just let them die down naturally.

If your anemones are not any of the ones I've mentioned, it would be helpful if you could post photos.

You say they are Japanese anemone of various colours. So if you have the id right they are shades of pink with white .
I leave mine until the stems are dead and then just cut them down to about 3inches above the soil. In fact I cut mine down last week.

I just spray them with weed killer! They are terrible thugs in my garden.
Otherwise do as Seaburngirl suggests.

Surely that's only the bog standard pink ones? Would you weedkiller Honorine Jobert?