JSLL #25 - Urashima Tarou - Twenty-sixth sentence

 The end of the Princess' utterance is sentence 26 of Urashima Tarou:

しかし、決してこの箱を開けないでくださいね。」

しかし、けっしてこのはこをあけないでくださいね。」

Shikashi, kesshite kono hako wo aketenaide kudasai ne."

Note the closing quotes at the end, to signal the end of Princess Oto's monologue.

しかし, which exists in kanji form but is almost never written that way, means "However."

決して looks like a te-form, but there's no verb that would have it as a te-form. When followed by a negative clause, it means "(not) ever, (not) by any means, (not) in the least, (not) in any way, (not) at all." As we'll find out, the clause that follows is negative.

決 is an N3 kanji, which occurs in 54 common words. Most have to do with deciding or decisions.

この箱, followed by the direct object particle を, is "this box."

開けないで is a negative te-form of 開ける, the transitive verb "to open." 

Te-forms actually have two negative forms, with slightly different meanings:

• The ～ないで ending is used for the negative imperative ("don't open") and to express "without" ("without opening").
• The other ending is ～なくて, and it's used to express a causal effect between the clause to the left ("Because someone didn't X, ...") and the clause to the right ("...Y happened.").

What follows this ～ないで form is ください ("please"), so this is a negative imperative.

The only thing still left is the ね particle at the end. This is a sentence-ending particle, which typically reads as something like "...wouldn't you agree?" or "...right?" In this case, I'd say it adds a bit more to the "please."

I would translate the entire sentence as:

However, please never open this box, okay?"