Diving around Dubai is not really that great, viz is generally at best .... well appalling 1-2M being the norm.
If you really want to dive from Dubai I would look at Pavilion Divers based out of Jumeirah Beach Hotel, they dive the wrecks in the Gulf, long boat rides though.
The better diving is over on the east coast of UAE, 2 hour drive from Dubai.
Al Boom run a cattle boat over there, with transport from Dubai to Al Aqqa and their base is out of the Meridian Hotel.
Divers Down run their east coast op from the Miramar Hotel
Great diving in the UAE - Divers Down UAE - Divers Down UAE - SCUBA Diving on the East Coast of the UAE in the Miramar Hotel
In my opinion they are a lot more professional than some other ops. If you want to self drive, you can try Scuba2000 about 5 mins drive south of Divers Down but don't expect any guided tours as they rarely have a DM on board, but some really good divers use this op
Scuba 2000 - Diving in Dubai - UAE. A dive resort close to khorfakkan, bidiya, fujeirah, al aqah, sharjah, Dubai
PM me if you want more info, I usually leave Dubai around 6.30am to drive over to the east coast most Fridays and occasionally on a Saturday if I don't dive on the Friday. During the week you could dive the two aquariums in Dubai but that is probably more expensive than driving over to the east coast for real diving.