All Battlepods (except maybe The Big Iron) repeat a certain pattern. This pattern is not always the same between missions, but once you figure out what moves they do and in what order, you can time killing the monsters before the shields break.

Another way that seems to work is to actually trigger the monster into attacking. This works especially well since monsters tend to move slower during their attack and have a cool down period after they attack. Luring the monster away from the Battlepod and then doing this will give you ample time (provided that it was done just before the Battlepod's shields break) to whack the Battlepod into oblivion. It isn't always advisable to do, as it takes some practice with learning monster patterns and is a bit risky.

You could also use that funky shield bash. It not only pushes mobs away a considerable distance, but it stuns them as well, giving them a harder time to reach and attack you after you run away.
Just be sure to shield bash when the mobs aren't charging up their attack.

Changing your harness from normal to elemental does increase the damage towards Battlepods, since they are considered Construct and are thus weak to elemental damage.

Getting an Elemental Harness for Seraphynx is a good idea, since Seraphynx's Ray of Light works very well against monsters that are immobile or slow, such as constructs, battlepods, and some undead; all of which are weak to elemental damage. However, it's 8,000 Crowns, so it is up to you whether you want to drop the money for it or not.

Edit: Oops, from what the wiki says, all damage is neutral to Battlepods. Although I could've sworn elemental does more.

If you have trouble with a mission don't be afraid to ask for help from other players. When I was a young knight I found that asking for the help of champions and vanguards was extremely useful. There may be a few who are rude to lower ranked players but just keep asking. Another way to get help is to ask your guild mates for help.

If you are still having trouble I found that offering a small reward to those you ask for help can get recruits for a mission faster. (promise a set amount and pay them at the last lift)

If you are playing with knights of your same rank then support your team my drawing the aggression of the monsters and just leading them in circles. If you plan on doing this make sure that everyone in the party understands that you WANT the monsters to chase you. As long as you are the person who has done the most damage to a monster (and are within a certain range) their simple processors will have only you on their mind.

I encourage your use of swords and guns. I've found that using the two together is highly effective, especially when using a sword with high-knockback. I typically will whack something across the room and then shoot it until I'm close enough to whack it again. Or, if my HP gets low, I'll usually stick with gunning. As a DVS (AKA Cutter) user, it's particularly important to carry a good sidearm, since the Cutter swords are somewhat risky and short ranged.

Anyway, another way to deal with constantly respawning enemies is to Freeze them. If you can get a Cryotech Alchemer (gun), a Frostburst Brandish (sword), or a Freezing Vaporizer (bomb), (Or anything else which you can use in T1 that freezes), you can use those to freeze foes rather than killing them. Once you get to stronger freezes (like that of the heavy freeze sword), you can leave things to chill for 6-8 seconds.

Note that I would have linked you to the Spiral Knights wiki, but as of posting this, the wiki is down. (Why...?)