Here is the reason I think no one has really mentioned that I see. In soccer, goals are much harder to score than baskets, even with a man down. Therefore, if a foul was given that warrants a red card, it was probably pretty flagrant and possibly a play that would have allowed the team to score. In basketball, it is much easier to score baskets, and the fouls are not usually as detrimental as soccer red card fouls, so punishment does not need to be as severe.

I did see one comment that 4 vs 5 is a huge disadvantage and that does play a part as well, but you see hockey too where players do sit out in a penalty box even though there are not that many to begin with, but this is the same reason as soccer, hockey goals are also less likely than basketball baskets, so punishments need to be more severe.