Every one of them should start with: "Space, the final frontier..." and in some way echo the new life and new civilizations motif, just to remind everyone out there this is supposed to be Star Trek.
DS9
Space, the final frontier. ~~It was the dawn of the Third Age of Mankind~~ On the edge of an unknown part of the galaxy stands Deep Space Nine: a place to explore new worlds, a home to welcome new life and new civilizations, and the first defence against the dangers to come.
VOY
Space, the final frontier. The USS Voyager is lost in the Delta Quadrant, 70 years from the Federation. Joining forces with our former enemies, the Maquis, we're doing everything we can to survive, to continue Starfleet's mission to seek our new life and new civilizations, and set a course for home.
ENT
Space, the final frontier. These are the voyages of the first starship Enterprise. Our mission: to explore strange new worlds, to seek out new life and new civilizations, to boldly go where no one has gone before.
DIS
Space, the final frontier. The galaxy is filled with wonder and danger, and the Discovery is ready to face both. With our ship and our crew, we will continue to find new worlds and seek out new civilizations, in this time or beyond. We are explorers. We are Starfleet.
PIC
Same as TNG. No other narration and no other narrator would feel right.
LD
Space, the final frontier... (Boimler, what are you doing?) These are the voyages of the starship Cerritos (Wait, are you seriously...) Its ongoing... (oh for the love of...) Will you stop interrupting me?! (Dude, we've got to scrub the carbon filters!) Sigh Fine! (Aw, come on, we can still boldly go after we've cleaned up everyone else's mess) That's all I wanted to say...
PRO
Space, the final frontier... the USS Protostar has been "borrowed" by these young people trying to find the Federation. As their holographic training advisor, it's my job to guide them through the galaxy, where they'll deal with new life and new civilizations, and boldly go where they've never gone before.