Active boosters simply multiply the number of skill points you receive at the end of each battle. As such, you are better off activating them when your pool of battle skill points is full (e.g. at the beginning of each weekly cycle). If you play a lot, you might be better off with a shorter (e.g. 3-day) active booster as it will allow you to earn bulk of the extra SP by playing a lot while it is active. If you prefer a more casual style of play, you might prefer to have the boost active throughout and have the freedom to spread your play over more days without worrying about it too much.

Now, in regard to the weekly skill cap and the implications of it:

We acknowledge that the current implementation of the skill cap still isn't quite there yet. It doesn't quite allow people to choose when to play as freely as we would like, and it also adds some unnecessary complexity when it comes to using active boosters. This is why we are working our way towards a rollover SP system: instead having your pool of bonus SP being reset according to a fixed weekly cycle, you will steadily accumulate bonus SP to a pool over time that you can then empty through fighting in battles. As a result, everyone should be able to better choose when to come in and claim the available SP. Moving to such a system would have the added benefit of allowing us to make active boosters to give out a constant value over time by having them influence the rate at which SP accumulates into the pool instead of just multiplying SP received at the end of each battle.