While all the talk is of visiting Ha Long Bay on an organised tour, it is sort of possible to visit it independently. Sure you'll still end up on a tourist boat, but you won't be a part of the group onboard. Is that close enough? We thought it was and here is how we did it.
Starting in Hanoi, we jumped in a taxi to Gia Lam bus station early on a Friday morning and upon arrival were accosted by someone asking if we were going to Halong City, so we followed him to the ticket office, bought a ticket for 120,000 VND and were plonked on a bus. Half an hour later the bus left for what turned out to be a trip of more than four hours (including a short rest stop). The bus wasn't too busy and was kept cool, so we were comfortable enough. Around six kilometres from our destination we were told to get off the bus and were immediately pounced on by a motorbike taxi driver offering a lift into Bai Chay for 30,000 VND. On we hopped.

One option is to start your Halong trip here.
We decided to spend a night in Halong City before heading out to the bay, so checked into one of the numerous budget hotels and spent the afternoon happily pottering around town and along the promenade. In the evening we ate delicious barbecue food, downed a cool Saigon beer and checked out the Marine Plaza.
The next day we wanted to go to Cat Ba Island but not all tour operators could get us there by junk most suggested travelling via Tuan Chau ferry. However, we found someone who charged 300,000 VND for the boat trip, including transport to the harbour, so went along with it. We were picked up at 12:00 and at the harbour we had to buy our own entrance ticket to the Bay (120,000 VND). Then we waited and waited. Eventually a guide approached us with a group just arrived from Hanoi and we followed him out to a boat.

Don't expect luxury on a day trip.
On board, our group of about 20 people was herded into the dining area for lunch. My ticket didn't include lunch so I escaped to the sun deck, although calling it that makes it sound much nicer than it was: a few battered and broken loungers and random chairs scattered around a deck that had seen much better days. Still, as the first one there I claimed the best-looking lounger and relaxed with my book and a Diet Coke, available on board for a reasonable price, as we started the journey into the Bay.

Join the kayaking or sunbathe on deck.
The boat made two stops en route to Cat Ba Island, the first at a cave and the second kayaking. My ticket included neither, but I could have paid an extra 100,000 VND to join the kayaking or bought a cave ticket back at the harbour. The guests who participated seemed happy enough.
After the kayaking the boat cruised the last stretch to Cat Ba Island. Everyone sat on the deck as the beautiful landscape rolled past and we hardly saw any other tourist boats, making it a peaceful and enjoyable hour. Once we arrived at Cat Ba though, the journey was marred by the long wait we had at the pier for the bus into town. It was around 19:00 by the time we got to Cat Ba town and checked into a hotel.

A long wait at the end of the day.
While we just overnighted on Cat Ba, staying there puts you nearer the attractive parts of the Bay, which is ideal for those who want to explore. It takes about an hour from Ha Long City to get out to the karst area, and as well as booking onto boat tours you can also explore the island and national park and book onto adventure activities such as rock climbing. The beaches on Cat Ba Island are also better than the beach in Ha Long City.

Peace and quiet.
Another option we could have taken was to take a bus to Halong City and buy a ticket for a round-trip boat tour at Bay Chai harbour. A four-hour trip will cost 100,000 VND plus the 120,000 VND entrance fee. Or pay an extra 50,000 VND plus cave entrance for a six-hour tour including a trip to the cave and beach. Boats leave at 12:30. Agents in town will also sell you tours for US$25 to $45, including transport to the pier. You'll probably need a night in Halong City afterwards as buses don't run too late.
Alternatively, you could buy a combined bus and boat ticket to Cat Ba Island from Luong Yen bus station and book boat trips out from the island.

No room for negotiation at the harbour.
Would we recommend a DIY trip to Ha Long Bay from Hanoi? If you only have a day no. Book on a budget tour from Hanoi because it'll cost about the same and be less hassle. Plus you're guaranteed lunch (but probably not dinner).
Neither is there much advantage in going it alone if you want to do a two-day, one-night trip with an overnight stay on a boat. You won't save any money by booking in Ha Long City or Cat Ba Island and you'll still essentially be on an organised tour you'll just have to sort out the bus yourself and leave Hanoi earlier to ensure you get to Ha Long City in time to book and get on the boat.

The views are the same whichever approach you take.
However, if you have a bit of time and want to explore Ha Long City and Cat Ba Island, then travelling independently is recommended. Do as we did, and spend time in both Ha Long City and Cat Ba, or travel direct to Cat Ba Island and make that your base. We like Ha Long City, but unless you have a particular desire to go there, we'd recommend heading straight to Cat Ba from Hanoi.

If you DIY you'll have time for the beach.