if have a 1 drop with boost 1 in hand and I recruited a doctor minerva will I be able to play my 1 drop with the boost for just 1 ressource point ?

no, boost increases the cost to recrute a character. when determining the cost of something, first you apply anything that reduces the cost (in this case "Press") then you apply anything that increases the cost ("Boost").

More to the point... Press can only reduce a recruit cost to a minimum of one. And boost is an added costs, that goes on top of the basic recruit cost. So you will never be able to pay less than one, and the boost will never be reduced by press.

Even if you brought the cost of a character down by 8 using some Franklin Richards or Alicia Masters trickery... you woudl still have to pay any boost costs in full, as they are a seperate costs on top of the recruit.