I don't lump all gun owners together.
I can speak to this as someone who will never carry a gun of any sort. It's not about the guns themselves. It's not about limiting anyone's constitutional rights. It's really about that singular word you used - responsibility. It's obvious the constitution doesn't spell out the details of gun ownership, but a smart society would, and does, put certain perameters on that ownership for the greater good. This is why we're not allowed to have an RPG hanging in our truck window next to the Sidewinder missile launcher. Same reason even the most staunch gun advocates don't think it appropriate certain individuals be allowed to carry. Someone with a defined mental instability shouldn't be in possession of a gun, and some of them are as American as Yanky Doodle himself. What about their constitutional rights, versus their ability to be responsible?
I'm not a fan of guns, but I'm a fan of an individual's LEGAL right to be able to carry one if it makes them feel safe, or even if they just want to. As a citizen on the street with those people, I'd just prefer they have proper training to responsibly carry and use that gun. Most gun owners I meet are very passionate about their rights to carry, and would easily pass any training or test with flying colors. The DMV checks your vision and driving acumen before you take to the streets in a feeble, but helpful, effort to make sure you're safe to drive. I'd like to know that gun owners are mandated to go through even the most basic training so they too are safe...er. Safer.
If nothing else, I'd prefer to see a strong and healthy gun community that can proudly say they not only have a given right to carry, but they've proven themselves capable to do so. That wouldn't leave much room to even threaten to take those guns away.