Since 1848 the Netherlands have been a constitutional monarchy. The head of state is the Monarch who in return is governed, so to speak, by the constitution. The constitution also dictates that the Monarch can not be held responsible for his or her actions, much like diplomatic immunity. The prime-minister holds the final responsibility for the Monarch and the Royal family. The Monarch signs all laws that pass the Dutch parliament and the Royal Decrees (these are mostly decorations but also mayors and commisioners).
Once a year, the third tuesday in september,on prinsjesdag ("princeday") the Monarch reads the "Troonrede" (the Royal Speech) which elaborates the governmental plans for the coming year. The influence of the Monarch is mostly noticable during the formation period of a new government. Due to the Dutch multiple party system, the government is practically always formed of several parties forming a majority in parliament. The individual that stimulates this coalition of parties is appointed by the crown.
The ministerial responsibility regards the whole Royal family , as mentioned earlier, but only when their public appearance might possibly affect the public interest. The Royal family, or the "Koninklijk Huis" as the Dutch call it, has to have permission for its public actions from the government. Possible heirs to the throne have to ask the permission of the government if they want to marry.