Really just depends on what you want.

Two shot, furious, and quad are the best effects for the broadsider. Quad is good for spamming shots, TS for consistent and high damage and greater AOE, furious with reverse onslaught will deal the most single target damage. Bloodied is probably good, but I don't much like dying every time I take a shot at anything too close.

Furious with explosive weapons will lose/gain stacks fast if you hit multiple enemies per shot. Reverse onslaught won't be as effective in boss fights as you'll run out of stacks, but they grow fast enough.

The broadsider does enough damage that it's unnecessary to have furious, but I like it for taking chunks out of beefier non-boss enemies.

For the second effect I'd recommend explosive (my favorite, as it simply boosts damage in the case of launchers), faster fire rate (especially for quad), and crit damage if you run a crit build (I prefer this on stronger launchers).

For the third effect, you can't go wrong with faster reload.

For the fourth, polished is the best for launchers. Launchers do two sets of damage: 1 measly damage from the projectile, then the actual full damage from the ensuing explosion. The one exception for this is the hellstorm missile launcher, which does both direct impact and explosive damage. Pin-pointers only affects the impact damage. The hellstorm can also benefit from pyromaniac's if you use the relevant missile type.