I just use scissors and a delicate touch (grab it by the corners, not the actual face of the strip), and I've never had any loss of potency that I could notice. As far as storage goes, I put it in a pill bottle, inside a baggie, wrapped in foil to keep light out, then just put it in a dark cool cabinet. But I usually take all my doses within a few weeks of getting them, so I can't comment on the long term efficacy of this method.
LSD isn't as unstable as people make it out to be. Just don't go keeping it in your hot, cramped wallet like so many people do.
Also, when I finish it off, I lick the foil it's wrapped in. Idk if anything is ever actually left on it, but it can't hurt.