If something is above board, it's been done in legal and honest way.
1) I'm sure the deal was completely above board as I know James well and he would never do anything illegal or corrupt.
2) The minister claimed all the appointments were above board and denied claims that some positions had been given to his friends.
3) The government officials are being investigated for corruption. If everything they have done had been above board, they will have nothing to worry about.
Possibly derived from card playing and the fact that card players who keep their hands above the table (or above "the board") can be seen to be playing honestly, without cheating.