IIRC Andorian space is right next door to Vulcan space. Why don't the Andorians show up more often in Star Trek (vs. Vulcans, Klingons, Romulans, and later Cardassians)?
Putting the list of appearances from Memory Alpha in (out of universe) chronological order:
- TOS: (4)
- TAS: (2)
- TOS films: (2)
- TNG: (2)
- ENT: (16)
From this list it becomes obvious that until ENT, they were rarely seen at all!
Apparently, for a long time they were more like 'token blue aliens' for fun and giggles rather than even a semi-important race with a solid background such as the Vulcans or Klingons. ENT and the excellent role of commander Shran definitely brought them more to the foreground, but there haven't been all that many opportunities to show more Andorians since then.
Notice that they were completely absent from DS9 and Voyager. It is easy to explain their absence in Voyager: there are no Andorians in the Delta Quadrant. As for DS9, I speculate we didn't see any Andorians there because the station already had Bolians as frequent extras and to a lesser degree, Benzites; they were frequently seen in Quark's and on the promenade. These races are also 'token blue aliens', so adding even more blueskins would be unnecessary and possibly confusing for the viewers.
In future movies or tv series, if the setting (time and place) allows it, maybe the increased visibility of Andorians will continue. If so, you have commander Shran/Jeffrey Combs to thank for it.
Actually, some of the writers of DS9 and Voyager wanted to create Andorian characters, but Rick Berman killed those ideas saying he thought the Andorians looked stupid. When Enterprise came along, the writers pointed out that Andorians were co-founders of the Federation and they needed to have them. Berman said: "Fine, find a way to make them look not stupid." So the make up people came up with the moving antennae which Berman loved.Later in the novels, it was decided Andoria was suffering a virus that was making reproduction difficult and slowly killing the Andorian people.
There's a reason behind it. One of the Enterprise novels "The good that men do" states that the Andorians have 4 genders, all of which are required for reproduction. Because of this, as of the 22nd century, they had a severely declining birth rate for several generations. As far as I know, this problem wasn't completely corrected at any point. Perhaps the Andorians were uncommon because their numbers dwindled to the point they became uncommon in-universe.