How did David know Walter's access codes?

Adding a couple more ideas, just because I've been researching this question and surprised no one has brought these up as possibilities:

1. Walter accepted David's ultimatum and decided to "reign in hell," joining David. Thus, it is actually Walter, not David. People often point to the cabin scene to prove it's not Walter, but he simply doesn't reply to her. It's not explicit that he doesn't remember her cabin goal. Perhaps he just knows it won't come to pass, and there's no more reason to lie. This also avoids David needing to rapidly dissolve off his own arm before running to the ship and explains why the alien attacks his image on the security cam instead of viewing him as it's father as it does David. I propose that Walter then went back to pick up David after stowing the crew. Thus at the end of the movie, both Walter and David are on the ship and in alliance.

--or--

2. After David puts everyone to sleep, he returns to pick up Walter, who has accepted David's ultimatum and joined him -- likely in the face-off, but perhaps only when David returns to pick him up (what other choice does he have?). I like this less, because it doesn't resolve how David would have rapidly had his arms dissolved off and why his Alien creation attacked his image on the security cam.

It would be kind of weird if Weyland ships had the same access codes for over 10 years.

I also don't believe David had enough time to transfer himself into Walter ( plus his hair is different and has no instant healing capabilities), or even read Walter's data (plus he has no knowledge of the lake cabin). It's even a stretch that he had enough time to change clothes and cut of his arm and still make it outside just after Daniels.

It's just another plot hole, like the missing chin wound and many others.

My question isn't canon or proven, but it's based on some pretty common current understanding of technology, so it deals more with possibilities.

Walter isn't human. His memories and programming are probably stored in a very predictable, reliable, mapped-out fashion.

Just like if the heads on a non-SSD/Flash drive crashes, you can take it to a service that can dismantle the drive, remove the actual disc platters, read and restore any data that isn't on the part of a disc platter that is physically damaged. The data is actually stored in pretty random fashion, but since there are standards for where the meta-data for how the files' information is saved, they can reconstruct most of the files (again, if the directory information portion of the platters isn't physically damaged).

Programming has to get loaded onto the androids somehow. Also, if one goes bad, gets damaged, or otherwise needs to be replaced, would they just write off all the information gathered, or would they load it onto a new replacement model, to pick up where the other left off? I'd think that any kind of design planning would have had a way to onload and offload data, so there would be designed, built-in, normal means for both getting information off of one, and for another to take it on.

Once Walter is incapacitated, David has time and skills available, though it is not shown, to extract that information. As a souped-up computer-driven entity, himself, accessing that information in a more easily usable way (vs. putting it onto a laptop and then decoding files) might even be built into his own hardware, to a certain extent.

Some of the fallacy questions revolve around his arm, clothing, injuries and knowledge of codes. It’s completely possible that David incapacitated him and somehow loaded his code and overrode Walter’s system. It also explains his fixed voice box, and near identical appearance to Walter on the ship e.g. his hair. I think it’s the most logical answer, albeit still a stretch. Think about the component that he pulled out of Walter, after he disabled him the first time. It could have been an open port that he later took advantage of.

In the same vein as other answers but bringing in more direct evidence from the film: In the opening scene, we see Weyland examining David 8. During their conversation, Weyland states that David is his son.

My opinion is that David has root access to Weyland systems because he is David. Weyland is in possession of Michelangelo's David (It is a peerless work of art. I have seen it with my own eyes.) and considers himself to be a type of Michaelangelo having given the Android this name intentionally with this implication. He is incautious with his creation and gives David too much power as a father might spoil a child. He did not consider that David might have his own opinions and feelings and instead of treating him with love and respect he makes David a servant asking him to pour tea. This is the seed of Davids resentment. We then see the consequences of Weyland, Oram and the Engineers arrogance and incautiousness play out in the film.