Mazarin didn't mind if Anne gave Henrietta a small pension and a place to live, but he refused to consider the idea of going to war in England, although Henrietta urged Anne and Mazarin to do so. He persuaded Anne that it wasn't in France's best interests to help Charles. Thanks to his inaction during the Thirty Years War, and his inability to manage his own domestic affairs, Charles had an incredibly low reputation on the Continent. Mazarin honestly believed there was nothing to be gained for France by helping him.
If Henrietta had been executed, I don't doubt that Anne would have been sad about it, but there would have been nothing for her to gain by going to war in England. The wars in Germany and Spain, on the other hand, allowed France to gobble up territory.