I recently finished a temporary assignment in Djibouti, where my wife and I were able to take advantage of some free time for diving. We used Dolphin Excursions, based out of Djibouti town. They are really the only diving operations in the country that operates primarily in English.
We went on a one day dive off Mousha Island to see how the operation ran, before signing up for 4 days on their liveaboard. We were impressed with the quality of equipment, pre-dive briefings & stress on safety. Dan, Allyson and Kirsten, the instructors, were all professional yet fun divers. While we weren't that impressed with Mousha Island, we felt better about going on the 96-hour trip to the Bay of Goubet.
If you have the opportunity, don't hesitate to sign up for their liveaboard. The entire trip ran smoothly, diving 4 times some days, with a good group of staff & divers. The ship captain was incredibly knowledgeable of the dive sites & gave clear, easy to follow briefings. Diving was from skiffs, and the local skiff pilots were very helpful in pickup & drop-off. And the food on board was great, an added bonus.
The other dive operation in Djibouti, Blue Lagoon (Lagon bleu), works out of Kempinski Hotel. I found them less than forthcoming about dive pricing, giving individual dive prices without telling us their trips require you to pay for meals as well. (They do not allow you to take food or drinks on their dives to Mousha Island, but force you to pay for meals through them.) Additionally, they majority of their clientele are French speaking, so most of the instruction/briefings are in French. But I was told by local divers that Lagon bleu is a safe company to dive with.
Warning on Djibouti: The country is incredibly expensive compared to similar countries/locations. The cheapest, clean hotel we found was $60/night (le Heron Auberge). Most will cost at least $120/night, and those in the center of town lacked a lot in basic cleanliness. Djibouti is not a true tourist destination, and businesspeople their do not know how to cater to tourists. You will struggle unless you have at least some basic French language skills. AND IT IS FREAKING HOT!!!
Safe diving & safe travels.