Pretty much limits you to a standard attack. Move action to boost by the listed damage, standard to fire.
What character can best make use of that?
ranged characters
Oh, jeez. Stunned by the insight at the core of your comment.
Maybe something more in depth. Feats or classes that work well with it maybe?
For example, I think Exocortex Mechanic might be a good option, after Combat Tracking is on.
First, let's consider Boost itself. It's a move action to increase the damage a weapon deals with its next attack that lasts until the end of your next turn, so you can Boost without attacking immediately. It's also optional, so you can forgo the damage increase and instead full attack.
Second, let's consider the weapons. Skimming through the weapons, there are only a handful of sets. The Aphelion and Parallax laser pistols are simply steps along the laser pistol progression that fit comfortably between the preceding and following options. Boosting gives them a bit of an edge, matching or almost matching the damage of the higher level option that replaces them. Sonic Pistols and Rifles alternate in their progression between Boost and normal weapons, and for both each Boost weapon is equal to or stronger than the preceding normal weapon even before its ability is considered. Unfortunately, the streetsweeper rifles have a noticeably worse ammunition consumption even before boosting, so that is a point against them. Finally, there are the Plasma Casters, which are functionally tangential to Plasma Rifles, but comparable to Laser Rifles. The two casters have comparable damage to laser rifles of the same level, and the option to boost them. Like the streetsweepers, they have worse ammunition consumption, and in this case they also have a worse damage type once resistances come into play - two is not better than one in this case.
So as a baseline, the trade-off for most boost weapons is higher potential damage for higher charge usage, and still more damage for even more charge usage. For best exploiting this, it simply becomes a question of whether any given situation would be better served by full attacking or getting a single stronger attack.
And for that, you really want the Envoy, Mechanic, or Technomancer. If an Envoy doesn't need to use their move action that round on one of their Improvisations, they can instead Boost and combine the attack with one of their other Improvisations. A Mechanic who has a Move action available can Boost and then attack with Overcharge - though at this point they are using the equivalent of five shots per shot. And a Technomancer abusing Spellshot will already be making single standard action attacks anyway.
The potential weakness there is that it's not clear that Boosting is still the best use of those move actions even when you're stationary. An Envoy can potentially be using that move action for another Improvisation each round. A Mechanic can just full attack if they're not directing their drone or locking onto a target with the exocortex. And a Technomancer might be better served with Empowered Weapons, which will give a bonus to hit as well as potentially greater damage so long as they can sacrifice a spell slot for it.
The Bombard Soldier can potentially benefit from this as well in conjunction with Heavy Fire, but since that's a full action, you're now taking the action over two turns. Sharpshooters run into the same problem, since their bonus only comes up on a full attack, and it only applies to the first anyway. Maybe a fringe benefit they can use when they end up having to reload...
Boosting may still be useful situationally. When you're faced with energy resistance you can't readily bypass, for instance, having a bit more damage per shot to brute force your way through may be more useful than multiple shots each getting reduced by the same amount. More rarely, Boost is going to be useful if you can set up an ambush or some other surprise attack.
I don't think Boost is enough to go out of your way to get it if there are better options on the table, but I think those are the ways you'd want to use it if your weapon happens to have it.