The strategies vary based on which expansions are being used and the number of players in the game. For purposes of this answer, I'll assuming you're using ONLY the base game and Daybreak (which is my favorite way to play BSG), and have a full complement of 7 players (including a Cylon Leader).
Overall, Battlestar is a resource management game. For the humans it means spending resources evenly as the game progresses. Resources are skill cards, fuel, food, morale, population, (lack of a) mutiny card, (lack of) enemy ships in space, (lack of) damage to Galactica, raptors, and vipers, nukes, etc.
Usually, if human are losing too often, it's because they're not using some of their resources properly. What does this mean? Well, after a loss take a look at what unconsumed resources you have. Maybe population is still in blue when you run out of food? In this case you should have spent more population (letting civilian ships die) to save cards for food checks. Maybe Galactica was undamaged at the end, in which case you should have spent less time shooting things in space. It may seem counter intuitive, but you have to let a lot of bad things happen (e.g. intentionally fail skill checks), to have the resources later when you need them.
You check/validate your resources by how far you've progressed on the jump track. So, based on jump track/jump distance you can evaluate where you 'should be' on resources. Fuel is an easy one to see, and can be used as a guide, as it tends to steadily drop throughout the game as you jump. The same should occur for other resources. If you're halfway through the game, and all of your resources aren't at half, then start consuming the higher ones to save lower ones. Remember, things like mutiny cards in your hands work the same way. If half of the players have mutiny cards halfway through, that's fine, don't spend them, but if everyone has one at the halfway mark, half of you should spend them to get back up. About 3/4 of the way through the game, let Galactica stay damaged, don't repair it, if you can avoid it.
As for Demitrius, the most important space is the Captain's Cabin. For one action you can get everyone a card. In a 7 player game, this is very powerful. It's even better to have someone sit there and be XO'ed to do it twice for effectively 1 action. Be careful about spending time searching the mission deck and running those missions. In a 7 player game (early), this is probably fine, as the threats haven't built up and you can get some extra distance, but actions are everything, so consider the opportunity cost of not spending them elsewhere.
For games with 3-5 players, I would consider ignoring Demitrius (including the cabin), it simply takes too long and with fewer players it's harder to hit the skill targets.
Carefully decide weather or not to play mutiny cards. They again cost you an action, and if your only purpose is to avoid the brig, reconsider if being in the brig is all that bad. You still draw cards, you still can XO (executive order), and the president can still use quorum cards, PLUS you don't draw crisis cards. This increases your effective skill-card-draws per crisis card draw. If it happens, consider leaving someone in the brig at least until people are starting to reach their skill card limits.
Things don't change too much for the non-Earth destinations.
- Don't over-pass your skills checks, this is a waste of resources.
- Intentionally fail at least 50% of skill checks.
- Make sure your Admiral is not a Cylon (most important)
- Make sure your President is not a Cylon (less important).
- BE Presidential, 1 resource for one action (e.g. Inspirational Speech) is VERY powerful.
- Don't waste too much time hunting for Cylons.
- Trust each other. If you don't, the Cylons have already won.
- Always XO.
- Advancing the jump track is more important than anything else
- Use the early FTL jump if you have extra population for that stage in the game.
- And most importantly, HAVE FUN!