I've actually found alot of our weapons very effective at killing BA...
Lasers - Sweeping motion lets you catch the BA if you're correcting your shots (even better the larger the laser). Tip: Chain fire if you have more than one laser
Pulse Lasers - High refire is really handy if you need to spray a bit to hit the BA. Again even better with the larger classes but beware of heat
Mguns - High refire and best spray and pray weapons against BA
RACs - much the same as MGuns, doing more dmg per hit
LBX - Who doesn't like shotguns?
Flamers - Flood the area with flames and I'm sure you'll take them out
PPCs - High splash area makes it really good at hitting the landed targets
Gauss - Kind of over kill against a BA, but with their high velocity you can find a blood splatter at what ever you precisely aim at
SRMs - Explosive Frag damage allows for flushing BA out of hide spots, unless they're already dead
AC2 and AC5 - High refire rate makes them really efficient much like the RACs and Mguns. Since they're higher damage, think of them like rapid fire 20mm or 50mm cannons
AC10 and AC20 - High Splash radius also makes these effective at killing the grounded BA from exhausted JJs. Again over kill, but it does the job. Much like a Broadsword to cut cheese
My main tip when fighting BA is that there's three major factors to consider - Speed, Accuracy, Friends.
Speed - Light and Medium Mechs should have little trouble when faced off against BA since you have the speed to get the range on the BA and then be able to shoot them as they try and close the distance. It's harder with Heavy Mechs, but still possible. Assaults have the worst time since they are slower and far less maneuverable.
Accuracy - Since BA thrive when zipping around with their JJ, don't bother to shoot them when you see the BA blast their Jets. Wait until their reach their apex of their Jump or when they get grounded from exhausting their fuel. You have about 1-2 seconds before they can thrust again which makes them vulnerable to your shot.
Friends - If neither of the upper two work for you, then your last resort (short of bailing out) is asking a team mate to get these pesky BA off you. Most often they will be able to have a clear shot on them and take them out for you