Are those single wall rims (whoops, my bad, I should have written double wall rims) with deep holes for spoke nipples? If so, thin rubber band type rim strips won't provide enough support. The tube will extrude into the spoke nipple hole and weaken, usually splitting just like the photo shows.

Try a cloth rim tape like Velox or other (they're all pretty much the same). Or a rigid plastic rim strip like Schwalbe sells. But the cloth tape works great.

The only trick with cloth tape is to avoid using tape that's too wide. If I climbs up the shoulders of the rim it can hinder seating the bead on some tire/rim combinations. I usually chose cloth tape just wide enough to cover the spoke nipple holes but no wider.