This is more for a blanket, scarf, or hat that is composed of more than 2 stripe colors. If you're only using 2 colors, use this method.
I'm making a blanket holding two strands of yarn together. Hopefully the pictures make everything clear enough.
Work the last stitch of the row up until the last step of the stitch. (single crochet in this instance). Cut your yarn, leaving a decent sized tail.
Bring in the new yarn color. Holding both colors together, finish the stitch of the last row. Be sure to leave a decent sized tail on the new yarn you're joining in.
At this point, I drape the tail of the old color and new color over the yarn I'm working with. Since I did this with single crochet, I chained 1. By draping the tails over and chaining around them it helps to anchor the yarn so it won't come unraveled when you wash and dry it.
After you've chained the appropriate amount, turn. Place the tails of the previous and current yarn colors on top of the next stitch. Insert hook and complete stitch as normal. Continue positioning the yarn tails over the stitches you're working into, and sc (or whatever stitch you're using) around them.
Holding all the yarn tails takes some practice but it's worth it. You definitely (at least in this case) could just tie a knot with the tails of the two colors. Just make sure you're always finishing the last stitch of the previous color with the new color.
Hope that makes sense! Have fun making stripey stuff. :)