Here is what you want to do to avoid that -
First, in early game, try to avoid fighting anyone or making enemies. If any civ asking you for money or tech, give them whatever they want. If any one ask you to allie against any other civ, try to take only the right to passage deal and drop the alliance part. With the right of passage deal on, you gain a huge bonus on the reaction from the other civ.
If you are going to declare war, make sure you 1) has no unit on the enemy terrority before declaring war, 2) are not in any trading deal with that civ, 3) are not having an active right of passage deal with that civ, and 4) are not in active peace treaty with that civ.
Basically you want to do whatever to avoid taking a reputation hit in early game. If you avoid conflict at all cost till middle age, chance are, all the AI will be fighting each other. At certain point, they will hate each other so much, they will not form an alliance against you. At that point, you can just focus one by one and take them out. I am not sure what size of map are you playing. But this works well on huge 15 AI civ map.
Another thing, if you are much stronger then the rest of AIs, they might focus on you. If you get to that point in early game, maybe you want to increase the difficulty level of the game.