B/X Mars ties race and class together into one package, as you might expect from the B/X designation. Here's how I'd map out their separation in 5e, without going down the road of bespoke races (though that is certainly an option I'd consider working on):
- Red Martian: Since "Red Martian" is the baseline race in B/X Mars, it makes sense to use either the "human" or "variant human" options from the 5e Player's Handbook for them throughout.
- Terran: Oddly, Terrans wouldn't use either of the human options. I'd make them goliaths to emphasize their strength, hardiness, ability to carry heavy loads, resilience, athletics, etc. It's not a perfect fit; in fact, I think replacing the goliath's mountainous trait with a d8 unarmed strike attack would be in order.
- Thark!: Tharks! can probably be replicated with half-orcs. That gives them an additional rage-like ability, higher strength and constitution, intimidation, etc. Unfortunately, it doesn't mechanize their extra arms.
- Red Martian Menton: Mentons are Mars's masters of the psionic arts. They're also the hardest to fit into the 5e rules, but the way I would go about it is to use the cleric class as the basis since mentons and clerics favor the same ability score (wisdom/psionics) and both use "spells" and can "turn undead." Making a custom spell list that focuses on enchantment spells, psychic damage, and illusion would go a long way toward getting the flavor right. I'd probably narrow down the domains to two choices: "life" and "death" correspond nicely to the alignments in B/X Mars.
- Red Martian Princess: Interestingly, the class that suits the princess best is rogue. The rogue specializations also give you a nice variety when it comes the role a princess might take. "Thief" works for crafty princesses, "assassin" works for sneaky princesses, "arcane trickster" is pretty good for princesses with psionic talent, "mastermind" works for princesses used to leadership, "swashbuckler" is a good fit for warrior princesses, etc.
- Red Martian Warrior: Red Martians are fighters. The "champion" and "battlemaster" sub-classes might be the only ones on offer, although "cavaliers" might make sense for mounted Red Martian Warriors. "Elritch knights" might make for a cool "psychic warrior" archetype, but I'd have to give that a second look.
- Terran: It is very tempted to make some sort of monk variant for Terrans to preserve their "they are very strong and punch stuff to death because of Martian gravity," but barring that they are fighters with the same range of options as Red Martian Warriors.
- Thark!: Tharks! are barbarians, and they're probably only ever berserker barbarians at that.
The DMG has stats for laser pistols and rifles that would work. Just replace "energy cell" with "tubes." Yeah, they'll be pretty deadly but in my experience B/X Mars has Gygaxian levels of lethality, so it's probably a good fit. You could always give the PCs a 10 hp kicker if you want to compensate.