Last I checked, it affected all XP you earn so long as you have the boost active. I believe it gets factored in to the mission selection screen.
Examples (from memory):
Boost Active: Admiral Quin offers you a mission. Mission reward already has XP boost factored in.
Boost Inactive: You start and complete a DOff assignment, but do not collect the reward. Activate Boost: XP recalculates to include boosted amount.
Boost Inactive: You start and complete several missions, but do not turn them in. Activate Boost: Missions recalculate the rewards at the time you turn them in, giving you the boost factored in to the reward.
If it wasn't clear you don't have to toggle on and off the boost. Its passive, and a new ui element in the top left can be configured to show how much boost you have left in xp and in commendation (doff xp, another type of booster).
Really good change over the older time-based system in my opinion, given you spend more time doing things that earn you low xp compared to handing-in or completing things.