Is it possible to mute some of my strings?

I think the most important things that I have learned during my years fiddling with a Guitar is to desperately work on honing the accuracy of your picking instead of going at the strings like a madman (if you'll pardon the expression :D). I often used to play with a heavy hand; any time I wasn't soloing, i.e. picking individual strings, I was hammering more strings than I needed and muting those that were unneeded with a combination of left and right hand trickery. I blame this anti-technique on learning to play on an acoustic first, where whacking out G/C/D chord progressions tempts you to sin against your right hand accuracy. Being self-taught, I had no one to correct me as I'm sure a good guitar teacher would do.

At times, playing fully muted strings in addition to un-muted strings is a very useful technique. For some great examples, see Where the Streets have no Name by U2 and pretty much any guitar work by The Edge or musicians in that similar genre. Some blues players like Stevie Ray Vaughn, would mute strings and "rake" in on the starting phrase of a solo to give it a scratchy, soulful sound. So it's not bad to play muted strings, it's just important to do it intentionally.

In conclusion, to fix this I would encourage you to work on your right hand accuracy by doing picking drills, practicing string skipping, and improving the overall precision of your right hand instead of putting a Band-Aid on the issue with capos and muting tricks. Focus on only hitting the strings you need to play the phrase. Everything from textural guitar to speed metal relies heavily on right hand accuracy if you want your playing to sound clean.