Beam weapons are the gnashing teeth of a good capital ship. 
When spying your ship, that's one of the first things your enemy will want to know - "how many beams does it have, and what are they?"

The beam weapons have the following major purposes for you:

Sweeping Mines:

 The more beam weapons you have on a ship, the more mines in a minefield you can destroy.  (Mine sweep mission)   ...and the higher tech the beams are, the more mines you can destroy.  You can see how these two factors can work for you. ;-)  I  seem to recall that the number of mines you can destroy with a given beam weapon are related to the square of the tech level.  Yes, that means that a tech 10 beam weapon (heavy phaser) will destroy 100 times more mines than a tech 1 beam weapon (laser).  Fly to the edge of the minefield to destroy it - don't fly to its centre! :)
 

Killing the crew of the other ship:

 Do you want to destroy the ship, or attempt to kill all the crew and capture the ship as your own?  If you capture it by killing all the crew, you don't have to tow it back to a starbase to claim it!  This is hard to do, though, if you're not the Privateers (who get a 3X rate on kill for beam weapons).  For doing this, you should use x-ray lasers, disrupters, or heavy disrupters.  Keep in mind that the latter two have pretty hefty explosive values, and you might blow up the ship before it is captured.  When debating whether to capture a ship or destroy it, check out that ships stats (in documentation) for how much of a crew it has.  Freighters have small crews, but Capital ships may have way more than you'll be able to easily kill.   If trying to capture a ship, you should combine these beams with Gamma Bombs, which have low explosive yield.  If you're daring, you can try and wear the other ship down with some *real* torps, first.   Careful - you may just accidentally blow it up, instead :-)
 

Killing fighters:

 Do those fighters come in and tear you a new asshole before you even got a shot off at the ship?  I hate that!  Before taking on that fighter ship, check the ratio of their fighter bays to your beam weapons.  In this case, have as many beam weapons as possible, as they can spew fighters about twice as fast as you can pick them off.  Any beam weapon will do equally well for this purpose.  Suicide ships with multiple beam weapons can deplete a fighter ship lickity-split, leaving it as easy pickings for the rest of your more valuable ships.
 

Destroying other ships:

 You may think that this part is straight-forward, and for the most part, you're right.  Big beams hurt ships more.  Grog smash puny ship.  However, beam weapons become less and less effective against ships with high mass.  When fighting the absolute largest ships, your heavy phasers aren't going to do any more damage than your lasers.  Remember that!  It will save you a *lot* of money and resources.   I saved this one for last because it's very important for long-term games.  You will throw many suicide squads at big ships just to wear them down.  Those suicide ships need be nothing bigger than a big hull with lots of lasers and a warp 1 engine.  (you drag the suicide ship into battle)  So, instead of spending 2000 megacredits on your suicide ship, you spend 200.  It's a lot easier on your colonists and their pocketbooks, whom you are probably already taxing to the brink of riot. ;-)