Significance of 門 in traditional chinese
On the radical 門/门

In both the traditional and simplified font script, 門/门 is a fairly common radical. Compare:


  閉/闭： shut (eyes...)
  
  間/间： interval, space...
  
  閑/闲： fence, barrier; idle 


In all these cases I find the door component very intuitive. There are of course plenty of cases where this is not the case (see the comment by user6065). The reason for your perceiving the radical as something predominantly traditional, may lie in cases such as:


  老闆/老板：boss; 


where characters were replaced by rather arbitrary other characters. There are also pairs such as:


  問/问;


which is classified under 口 in traditional Chinese and under 门 in the PRC.

The article which user6065 posted is about explaining the usage of 門 in some characters by its ancient phonetic value. Let us see, whether he will explain it himself, before I offer my amateur's perspective.

On 開心/开心

I am no native speaker, but 開心 is composed of the characters "open" and "heart". That "open" should feature a door or gate can hardly surprise. That the combination of these words means happiness is intuitive too. In both my native languages an analogous expression exists.

Greek:
Άνοιξε η καρδιά μου!
German:
Mir ging das Herz auf!