The pressurization in the cabin requires the doors have their seals working properly. How do they prevent pressurized air from leaking from the doors?
How do they prevent pressurized air to leak from the doors?
They don't. The seal does not have to be perfectly airtight.
The requirement is that it will create enough impediment to the air that wants to gush out, where "enough" is defined by the capability of the conditioning system to input new air into the cabin.
In other words, everything is fine as long as the air mass that gets out of the doors is less than the maximum air flow that the air conditioning system can input into the cabin.
Note also that, as stated on Wikipedia, aircraft are equipped with one (or more) "out-flow valve(s)" and "pressure relief valves": these valves intentionally let air out of the cabin to avoid over pressure in the cabin and thus they prevent possibly dangerous scenarios.
Like any pressure seal they are generally rubber seals. You can find one of Boeings patents here which covers door seals. Some smaller planes (and unpressurized GA planes) use inflatable seals, the door is closed and then the rubber boot is inflated to form a tighter seal. You can find an interesting discussion on the seals here but in general they may have tiny leak issues. This was the cause of at least one A380 diversion recently, but it looks like airbus is working through it and has a solution. Since the cabin has an outflow valve to keep it properly pressurized a tiny leak in the door will be compensated by the outflow valve. This airbus briefing warns that they can be hazardous and hard to detect if one does occur the crew may not know until the O2 masks deploy.
Slow/Insidious decompression involves a very gradual decrease in cabin pressure. Slow decompression may be the result of a faulty door seal, a malfunction in the pressurization system, or a cracked window.
Slow decompression may not always be obvious. The cabin crew may not notice the changes in the cabin, until the oxygen masks drop down from the Passenger Service Units (PSUs).
It has been noted that leaky seals on both pressurized and unpressurized doors can lead to a very loud noise in the cabin.
The seals do at least to some level try to keep water out. This is mainly because excess water may build up and then freeze as the airplane climbs. This ice build up in the door could in theory cause an issue (although I cant find any cases where it has).
A lot of aircraft with pressurised cabins have plug type doors which press against the fuselage when in closed position. There is a (silicone) rubber seal where it touches against the fuselage, which minimizes air leak. The image below shows the seal in an aircraft door.
Image from aviationtroubleshooting.blogspot.in