Question: One of the main problems I had to deal with, when I started to meditate and study the Dhamma, was the consequence of giving up attachment to reputation and pride.

Of course this is an extremelly positive thing to do, you simply become stronger when you stop caring about how other people see you, you just care about your actions, about karma, about your thoughts.

The problem is that people think you just don't care about anything else, they think you are not commited to your job anymore, to your appearance, to other people feelings... for instance: If they give you a bad feedback and you don't suffer and don't react to it, they think you are ignoring them.

It looks like we need small portions of drama to live in this society, non attachment to reputation makes you look like an alien or a brain washed person. I don't know how to balance that sometimes, any advise?

Answer: 
I don't exactly have an answer or advice to this, as I have a similar experience to a certain extent, but I understand that even though others might see the external us and base an opinion on it, the mind development that's going on for Buddhists may always be misunderstood by those not familiar with it and thus, dukkha arises for the one making the judgement. Maybe it also arises in us too, if we react to it even just a little bit. It's hard to be unaffected, I find. I guess that depends on where each one of is in the practice. As the Buddha describes, it is 'Patisothagami' or "it is a path that goes against the stream", so we're bound to hit resistance both from the stream and the other fish...to use a popular analogy. 

Maybe here equanimity is needed with compassion in order to keep us within the sphere of 'normal' worldliness whilst striving for something beyond it.

I hope the reflection is useful and thank you for bringing it up.

Best wishes

Paul
Maybe don't try to balance anything, just experience. Wanting things to be a certain way is always dukkha.
 Ideally, we should do what is appropriate to the situation(this is easy for Arahants, hard for beginners). So if they think they are being ignored, maybe use the misunderstanding as an opportunity to tell them about what your practicing. Say something like: "I'm not ignoring you, I'm practicing the Dharma and to stay in line with the Dharma I keep my words few but just right, it's called Right Speech. All Dharma practices aim at making the practitioner quit ignoring what makes themselves and everyone suffer, in order to transcend suffering and be happy. So, now can you start to see that I'm very far from ignoring you?"