What is a synonym for "conscious of reputation"?

Solution 1:

It's a compound phrase, but image-conscious:

concerned about the way one comes across to other people and the impression one creates
- Collins English Dictionary

It works for organizations as well as individuals. Of course, this has the negative connotation of being more interested in appearances than realities. If you mean the opposite, then you want conscientious:

very careful about doing what you are supposed to do : concerned with doing something correctly
- Merriam-Webster

Solution 2:

If you mean these companies are unwilling to take risks and they play it safe, then...

I'd call these companies circumspect

Wary and unwilling to take risks

the officials were very circumspect in their statements

or guarded.

Cautious and having possible reservations:

he has given a guarded welcome to the idea