Blast bombers are powerful utility knights, but if you don't know the functions of a blast bomb or the person bombing you could end up dead.
The first thing you should do is observe them to see what their bombing intentions are. Blast bombs can be used to create walls, to move enemies around, or just to simply create non monster areas.
Your basic Sword mindset says go after the nearest enemy but when it comes to playing with a bomber the nearest enemy could end up on the other side of the room in a second. So,what do you do? Obviously you go to where the enemy will be, this requires high awareness of the enemies in the room, keep your eyes on whatever walks into the blast range and be prepared to pick it off when it flies back (possibly stunned and served on a silver platter for you and your swords).
Gunners are a bit less affected, opposite of the Sword mindset you want to get closer to your blast bomber. Why? Lets think, gunners are ranged fighters, blast bombers keep enemies at a distance. This works to your advantage and theirs, you protect them from ranged enemies that plague bombers and they keep enemies at a safe range for you.
Bombing with Blast bombers requires a level of skill rarely seen these days. Depending on their playstyle you can either supplement their damage or defense. I can't really comment much since synchronized bombers require a level of communication bordering on telepathic.
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Blast bombs are basically any weapon you can alchemize from the 2 star blast bomb. All of these weapons have a bright flash that can make aiming at things in the blast difficult (as if the knockback didn't do that) and a reliable knockback (away from the center point). Understanding the amount of knockback as well as how enemy placement affects the direction are KEY to playing with a bomber.
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Of course my best advice is to start bombing yourself and learn to do it well.
Edit: Yes they do damage but so do all weapons.
you forgot about blast bomb most basic function. It does damage.