If a person is known to have conflicts of interest, or has been known to take advantage of conflicts of interest in the past, they shouldn't be given a major government position. Congress needs to take a good hard look at each of Trump's picks and ensure that they are suitable for the jobs they have been appointed to.
Many, if not all politicians, enter into their field to prevent, protect or further at least one particular cause (likely one that somehow touches them personally). A politicians entire career, in fact, is based on supporting legislature that matches with their (public and hopefully private) ideals. Like it or not, money is a gigantic part of everyone's personal life and ideals just like jobs and healthcare. If you do it to Trump's cabinet, you'd have to do it to everyone, and that's just unconscionable.
It is very important for members of the president's administration to avoid conflicts of interest, and it appears that Trump has not been careful with his picks in this regard. Therefore, it falls on congress to make sure that the people he appoints are going to do their jobs honestly and put the well-being of the country ahead of their own interests.
Donald Trump must divorce his business successes if he is to be trusted at all by the American people. For this reason, any conflicts of interest between Donald Trump the former businessman and Donald Trump the president elect must be viewed with a scrutinizing eye. America's well being is already tainted with too much private interest from big businesses.