In French, we also say meme, the word même in French means “the same” in the internet meme context, because the memes are always the same, with the textual ones exception.

Looking in urban dictionnaries, dank means high quality, but with internet memes, dank is kind of ironical.

I would translate it by Meme de grande qualité or Meme de haute qualité. Emphasis on the irony used here, because dank memes are willingly ridiculous and/or bad.

Note : In my city we also say “dank meme”, but although we speak French, we’re in the southern part of Belgium.

“Dank” might be translated by qui déchire ou qui défonce bien (like “dank weed”.)

The French translation for “meme” is the neologism mème (also oqlf), with a grave accent not to confuse with même = same, which has a circumflex.

However, this word is largely unknown in France so I would advise to add an explanation about it. To translate “Dank memes”, I would then venture:

[…] des mèmes¹ qui tuent […]
…

¹ Sur Internet, reprises virales et détournements humoristiques d’une expression, une photo, etc.

I may not share this opinion with everyone, but I generally don’t translate when it comes from the internet, especially memes. With my friends we say “dank memes” in French, and we all understand. We get them from English sources anyway so we don’t mind.

But if you’re trying to explain it to him I guess you’ll have to use French. I’d also say something like “des memes qui défoncent“, or “des memes de qualité” if you’re mimicking a posh connoisseur.

In addition to the propositions, one may translate it to “Mèmes qui (en) jettent” or “Mèmes complètement j’tés”.

It’s another topic, but French medias try to impose /mème/ pronunciation, like “même word”, whereas youth would more frequently if not always use the english pronuncition /mimze/ for memes.