They're extremely fragile. In fact, they're stored pressurized because laying on their side would damage them without the pressure keeping the walls rigid.
My understanding is that they don't need pressurization while being stored, but do need it to withstand flight loads. This is in contrast to, for example, the older Atlas rockets, which had to be pressurized constantly and would collapse otherwise.
Of course I can't find any authoritative source on this, although there are a lot of random internet folks like myself repeating the claim. And I guess I'm throwing one more on the pile here.
You're correct - the Atlas (ICBM as well as the current Centaur second stage) design that requires constant pressurization is known as a balloon tank.[0] SpaceX doesn't use balloon tanks, but it does pressurize tanks with nitrogen during transport to keep the tank walls rigid and prevent damage.
I unfortunately can't find an authoritative source, either, but I recall a /r/SpaceX discussion where it was stated that hurricane prep includes maintaining power because they're (lightly) pressurized at all times.
I wonder if they might maintain pressurization to increase strength in the event of some mishap, but it's not an instantaneous disaster if pressurization is lost.
I recall reading that they have equipment to maintain pressurization while the boosters are trucked to the launch site as well, so it's not just flight loads, in any case.
Yeah, they aren't balloon tanks (which absolutely require pressurization), but they do keep them pressurized as it improves their structural stability.
They're pressurized with nitrogen generally, just to keep them clean and dry inside. They're quite capable of standing or lying down while unpressurized though.
They're extremely fragile. In fact, they're stored pressurized because laying on their side would damage them without the pressure keeping the walls rigid.