The best way to cross the Creek, and probably the only one unless you want to make all your way to one of the two bridges that cross it, is by Abra.
These are small motorboat that make constant journeys from one bank to the other of the Creek.
There are several Abra routes:
-Bank Baroda to Deira Old Souq
-Dubai Old Souq to the Sabkha abra station on Deira's side.
They are a bit like buses. They wait for people to fill the boat and once they are fully booked they sail to the other side of the Creek. The price is very reasonable, 50 fils only and they will ask you for the money once you are on board. No ticket is required.
It is a bit of an adventure as you never know how good the abra driver is :-) we got a couple of them who were proper kamikazes :) good fun and a good way of mingling with local people and to get a flavour of life in Dubai.
Only a word of advice. The only women I ever saw on Abras were tourists. Don't go too exposed or you will have all the men looking at you. Very inconfortable I suppose...
Getting around in Dubai is pretty easy. It's a smallish city and there are plenty of taxis and buses. But..there's traffic...specially on weekends and throughout the Dubai shopping festival. The two bridges (Maktoum and Garhoud) that run across the creek and connect Deira and Bur Dubai are always busy. Same goes for the Shindaga tunnel.
The best way to get to Deira from Bur Dubai and vice versa, is by Abra. Abras are wooden boats that run on diesel. The fare is 50 fils (half a dirham). You can take an abra from the Al Seef walkway or near the HSBC bank building in Bur Dubai. However and abra may not always turn out to be convenient as they operate from only two points on the creek.
You can also hire an abra for a an hour or two. Its a nice experience. Shouldn't cost more than Dhs. 50 an hour.
There are three ways to cross the creek between Deira and Bur Dubai districts:
(1) By Abra
(2) Rashid Al Maktoum Bridge
(3) Al-Shandagha Tunnel
The cheapest of these is by Abra which is small boat very popular among locals to cross the creek. There is regular Abra traffic between Bur Dubai and Deira. At both sides they have Abra stations and you can enjoy the ride by paying just 50 Fils (appox one cent).
It was fun crossing the creek by Abra and I would strongly recommend you to try it.
The traditional way of getting across the Creek between Deira and Bur Dubai has been on an abra , a boat that's about 28 feet long and which can transport a couple dozen people. You board at stations on either side of the waterway and sit on what is effectively an unpadded dais atop the boat. Since the Creek is so well protected, it's rarely rough enough to cause anyone to worry about falling overboard even though there are no guard rails. The crossing takes about 5 minutes -- depending on the destination landing -- and costs 50 fils, or half a dirham, which is around 15 cents US. During slack times of day, these same boats will take you on 2 hour cruises for under $5.
A nice activity is to take an abra (a small passenger motorboat taxi) from one side of the creek to the other. The trip should cost around 50 fils. You can also take a dhow tour on the creek.
A fun and functional way to see Dubai Creek is from the small water taxis called Abras. It will take you from Deira to Bur Dubai side or other way around for only 0.50 durhams.
The Abra journey across the Creek is quick (costs 50 fils) and is how the locals cross from one side to the other.
Dubai is split in two by the Creek. If you want to cross it, obviously you can take a car to the other side but it's much nicer to pay 1/2 dhiram and use an abra.