I'm doing the Home Base mission and I have to bring a certain element to one of the builders. This element is usually found on barren planets. I don't want to waste a bunch of fuel in the hopes that I'll find a random barren planet. Will the color of a system like red ones tend to have more barren planets?

AFAIK, the spectral class (F & G (yellow), K & M (red), E (green), B & O (blue)) does correlate, at least weakly, with the richness of resources present in the system. I've seen speculation that the second character relates to the star's heat range within that class, from 0 the hottest to 9 the coolest. I'm not aware of an established relationship between the star type and the associated planets: lush vs barren or desert versus ocean.

I should have been keeping a detailed log, all this time. ;-)

Barren planets are AFAIK the most common ones in all systems, so you are really not going to struggle finding one. Just fly in a system with a lot of planets and moons and you are almost guaranteed to find one. Yellow stars will have more barren planets (on average).

But I am curious: what could you possibly find on a barren planet that you wouldn't find elsewhere? I guess nothing and I'm playing 1000+ hours...
Barren planets have only the most basic ressources, Platinum and Thamium which you can easily find elsewhere too. They hardly have anything more (I know but I wouldn't swear it), except those marked with 'hostile' sentinels: they protect some rare ressources, if you are looking for these, then chose ANY planet having hostile sentinels because you'll also find it there.

For the star colors:

Yellow Stars
Spectral Class: G or F
Warp Reactor Requirement: N/A
Yellow Star Systems are the default Star Systems of No Man’s Sky.
Common to find barren planets.
Basic resources.
Black Holes / Path to the Center waypoints only lead to Yellow Star Systems. (Never Red, Green, or Blue)

Red Stars
Spectral Class: K or M
Warp Reactor Requirement: Sigma Upgrade
Slightly more likely to discover flora and fauna on planets.
Slightly more rare resources will spawn.
Not accessible through Black Holes / Path to the Center waypoints.

Green Stars
Spectral Class: E
Warp Reactor Requirement: Tau Upgrade
More likely to find tropical / lush planets with more flora and fauna.
More likely to find planets rich with rare resources.
Not accessible through Black Holes / Path to the Center waypoints.

Blue Stars
Spectral Class: B or O
Warp Reactor Requirement: Theta Upgrade
Much more likely to find lush planets with plentiful flora and fauna.
Much more likely to find the the rarest resources.
Not accessible through Black Holes / Path to the Center waypoints.