I've been a GM for about three years now. I've been GMing for this group for almost two years next month. I have an issue with motivating my players to take action.
It's a sandbox campaign and they're currently going after the BBEG's old lairs to get clues on who he is and how he was defeated before so they can defeat him. The players have somehow got it in their heads that the BBEG is out to kill them; this is not the case because if it was the BBEG would just kill them and there wouldn't really be anything they could do about it, hence the fact that they're searching old lairs for ways to defeat him.
So I have tried laying out side quests where they get to be heroes, which is something the group has told me they wanted to be, to help them power up and gain experience, they even have mythic ranks. However whenever I present something to them that isn't just a straight up monster hunt or a murder mystery, they go "No way are we getting involved, it isn't our problem." The only way I've managed to get them to do anything heroic so far is to use the paladin's code to get the paladin to try and help people and save lives. On top of that I create villains and bad guys for the party to go up against, setting them up for awesome fights, and they retreat thinking its too dangerous, and once again, it isn't their problem.
I have tried using money to pay them, I have tried using their character motivations, but they seem dead set on ignoring everything in the world outside of the "main quest", in a sandbox campaign due to something that has not come up once since its introduction and there has been no in game reinforcement of.
If it helps, the party is level 5 mythic rank 2.
TL;DR: What method can I employ to get my heroic minded, but not heroically inclined, group of players to explore more of the side content so they don't get wrecked at the fight with the BBEG?
Best Answer
Everything is okay
In the question title you claim that you have a problem having the players do stuff; this is clearly not the case, as they are doing precisely what they should be doing in a sandbox game: They have set an objective and try to accomplish it, while ignoring unrelated hooks. This is good and effective play; I am happy when my players manage to create such a coherent goal and start pursuing it.
Their plan is to check the old lairs of the big evil thing, which seems, given the information you have given, an eminently reasonable thing to do.
You seem to not quite have adjusted to running a sandbox. In a sandbox game, the players take the lead. They are doing that. As a game master, you create the sandbox and content there. You are doing that. Players might have said that they want to be heroic, but play reveals their true priorities - either they see the big quest as heroic, or being heroic is not so important, or they see the big evil as too large a priority to focus on other stuff.
Whatever the case, if they players are not complaining, you seem to have a functional sandbox game going on. Prepare the sandbox in the direction they are going and continue as you are doing. Especially prepare the old lairs of the big villain, if any exist, and their surroundings, information about them, and the ways to get into them.
In my experience, the common problem with a sandbox game is that players have problems focusing and do bit of this and a bit of that, and then wonder why nothing is happening, while they only go to a dungeon here or there once and retreat and then go do something completely different.
Remember consequences
You write that there have been several situations where players have walked away from heroic opportunities. Remember the consequences of these decisions and develop the world according to their consequences.
Do not do this as a punishment for decisions made and paths not walked, but rather, as natural consequences of their action or inaction. In my experience, this creates interesting play, because players face choices which have consequences and they have to decide what to do, given limited time and options.
What if they fail?
In a sandbox game, the players might bite off too much and fail. If they attack an unknown enemy with no good plan for escape, then that is what they deserve. Start a new game some months or years into the future in the new post-apocalypse.
But this does not seem to be what they are doing, since they are explicitly looking for clues and information. Why are you afraid that they would go against the enemy unprepared?
Generally speaking
If you want a sandbox game with a more leisurely pace, then do not have a BBEG and certainly have nothing like a main quest. Rather, have several minor and major forces gathering strength and trying to succeed at their objectives, while putting the player characters in the middle of things happening. Then let them decide what to do.