In order to deal with it the first step is definitely to learn self monitoring, so that we can become aware of our emotional state and common reactions. There is plenty of psychological knowledge in this area, but it is IMHO rather disjointed between different fields and hard to get a comprehensive overview. Once we understand our reactions, it becomes easier (but not always easy) to control them, by learning new cognitive patterns.
For example, when I was younger I could become extremely angry. Eventually I understood that my rage didn't solve any problems and was in the end futile - real changes could only be brought about in a rational state of mind, the anger didn't help at all, it was just a wenting of frustration. So I began to learn how to control my anger, by slowly turning it into rational action instead (if something don't work, fix it instead of shouting; if somebody misbehaves, avoid him or find a way to change his behavior). Today I have very low aggresiveness levels (almost too low, I have been thinking of increasing them slightly), instead turning my energy in constructive directions.
So the method works, but we need better ways of implementing this form of cognitive modifications (right now psychology is far to illness-centered, and doesn't give enough help to make people more well; if this could be done, then it is likely that the prevalence of many bad mental states would decrease in society since almost everybody would be acting as good examples to everybody else).
I'm not sure how it could be improved. Increasing connections to the prefrontal cortex might be desirable, putting more of our emotions under conscious influence (the prefrontal cortex seems to be inhibiting a lot of our instinctive reactions; too much connections might be suboptimal too). There might be genes for that. Then there is the serotonin system; low serotonin turnover increases aggression and risk-taking behavior, which might be bad in this context (but we still want to keep our ambition while removing fight-or-flight thinking!). One explanation is that serotonin inhibits actions with unfavorable outcomes.
As for anxiety in general, that seems to be partially amygdala but also related systems such as the bed nucleus of stria terminalis ("the extended amygdala"). It is connected to the amygdala through CCK neurons, which suggests that if we could add some inhibitory neurons to them we could regulate anxiety better (CCK antagonists are a bad idea, since it is used so widely in the body). Overall, I think anxiety in general is bad and we can do well without it - it doesn't do anything constructive, and just decreases our energy, makes us afraid of socializing with each other, distracts from finding solutions to our problems and fills us with unhealthy corticosteroids.
Overall, I think that the optimal neurological solution would be if we could increase the connections from the prefrontal cortex to the amygdala, putting many of our reactions under conscious control. It wouldn't abolish them, and many are too quick to efficiently inhibit (we don't want to slow them down either - sometimes it is very useful to duck quickly or shift into survival mode), but we might get better control over our complex fight-or-flight programs, and that would be very useful.
Perhaps this could be implemented using genetic modifications, adding neuron populations to the prefrontal cortex and the amygdala that send axons to each other; far beyond what we can do when this is written, but given the huge advances in neurotrophic factors and development taking place today, this might get feasible in the near future. Another solution, more technological and less elegant, would be to implant suitably modified neural precursor cells into the areas and let them grow. This would help adults too, but it is hard to tell how useful this would be.
To sum up, the fight or flight response is basic and important, but hardly something we should accept in its current form.