[Dubious]

"Fake ID" is your answer. Possibly augmented by "Publicizer" (haven't used it). Check your Race Relations before docking and after completing a quest and/or monitor closely those log messages. I'm still a novice at this, playing "Normal mode" but I just hit sector 100 and have been questing for everyone (even those who Hate me), except for Pirates.

"Fake ID" lets you dock at any station regardless of their attitude towards you. If you are operating in a sector crowded with ships of Races less than "Warm", use it or "Stealth Coating" or "Cloak" to reduce the chance of them going after you. I've been able fly right next to "Hate" attitude ships with "Stealth Coating" and had them ignore me.

I am not sure if "Fake ID" actually does anything anywhere other than when attempting to dock. But it doesn't seem to hurt, and (so far) using it when it seems logical to do so appears to let me seem to avoid negative attitude issues. I've hunted Pirates for the Sigurn with "Fake ID" and "Less Than Lethal Weapons" (LTLW), and their attitude has dropped from "Loving" down to "Warm" but remained there through several sectors and hunts. (Suspect that is more due to LTLW.)

Also, I tend to try to alternate quests between "unhappy about" factions. Seems that avoiding pursuing a single quest line too exclusively eases tension. I've passed on four "end the game" opportunities so far (the Tenaculon, Vordalene, and Firaxian quests, and the "Amulet of Yendor") and have only two races that "Hate" me: Bankers and Monks (only one ship encountered and they Hated me on sight). When I start to run into the Monks more often I'll give "Publicizer" a try. (I've already decided the Bankers defeat will be my "chosen victory" for this game.) Currently the Firaxians are "Loving" despite the "all but last quest" for those opposing factions.

I do expect that to be more difficult at higher difficulty games.

-Dubious- "The object of the game is to enjoy the journey, not win the victory."