I built a simple base with a base platform, a room, a solar panel, and a hatch. The base seems to be powered, but it is full of water. (the same water level as outside the base.) How do you fill it with air?
@Mark Ripley's answer has all the suggestions I was going to make.
It's possible this is still a bug: https://trello.com/c/xHLuc8cM/1443-23037-water-surface-effect-inside-your-base-when-built-on-sea-level
Even though another answer was the solution, I wanted to post since there is another possible reason why your base is full of water -- you have a breach. When you add onto your base, you should get a little message about hull integrity or strength or something. The lower that number is, the more likely you can get a rupture in your base, causing water to spill in. It fills up in relatively real time, so if you're gone for a while, the base could be completely full, or if it ruptures while you're there, you might see what looks like an inch of water or so on the floor. I keep a charged welder with me at all times for these cases, and it's happened several times so far.
You seem to have the basics down, but be aware that the level of power delivered by a solar panel is dependent on light levels (night/day and ocean depth), and if your base is underpowered it will stop making oxygen/air. You might want to install a more reliable power source at your base.
Also, the distance (between power source and base) can cause problems, so best to avoid this problem by building your power source directly onto your base foundation.