You've become conscious of the feelings, now you have to have the right relationship with them. By being mindful, you can put them to a halt before they take over. It's OK to notice, but unconscious reaction can be put to a stop through being mindful of thoughts and feelings. And I know, it's easier said than done, but it gets easier and more instinctive with practice.
A very good answer. Thanks. I also agree with Dazino.
Feeling other's state-of-mind is a spiritual power (siddhi), which is a by-product of your spiritual progress. With power comes responsibility (sorry I had to say this

). Your responsibility lies in acknowledging that feeling and unconditionally accepting it without being affected by it. If you feel more adventurous you can try to help that person and show compassion. This is beyond the abilities of a normal person to do this (including me), so the best action is to remain unaffected and avoid the situation in future.
If someone is sending you their suffering and you remain calm or avoid them, you may be perceived as indifferent or cold hearted, which is fine no big issue. If someone is attacking you and dominating you and you avoid him, you will be perceived as weak, big problem, isn't it? You must deal with it using 'common sense', just don't harm that fellow too much

, and forgive after you are done.
It is unfortunate and bitter truth that in the modern world you start encountering negativity as soon as you step out of your house. For some poor chaps it starts as soon as they get out of their room. It can be a family member or even your spouse. For a Buddhist monk this is a perfect playground for practice, but an ordinary person who has become sensitive to energies finds it very discomforting. When they find that the negativity is omnipresent and making them sick, they give up the path they are walking, only the strong ones continue.
Initially , avoiding others helps a lot. If its a stranger or office buddy, its easy ... just mind your own business and don't see them much. Its a problem when its your relative or partner. If it becomes unbearable and long term, its better to part. (Please think before you do so, there can be other ways).
Other way is to build a positive shield around you. I frankly don't know how to do that. I've always taken the road of avoidance, even engaging in 'false-fights' so that the person stops talking to me. (I don't recommend a real fight or harming the person). Seeing more positive people and reading and watching positive an uplifting stuff helped me a lot. I've gone into the path of avoidance so much so that I have no TV and do not read newspaper, do not gossip, do not participate is those sick games people play with each other.......... and a long list of do's and don'ts.