I've tried patches and seam tape.  They don't hold long.  I use waxed thread and just sew it up by hand.  I over lap the fabric so that if water runs down it, it goes can't run into the seam.  Not the prettiest job in the world, but it works and I don't have to replace blankets every year. 

 Oh, I also spray all the blankets with a water repellent.  Not that we get much rain here, but it does get foggy and misty.  It helps keep the lining dry.

If the blanket is nylon or polyester - an iron-on patch probably WILL NOT WORK as the heat from the iron would most likely melt the fabric, the insulation (if any), and lining unless the blanket is canvas.

Attempting to sew, by hand, an iron-on patch will be another nightmare. 

Everyplace that you push a needle through will create a hole for rain to come through and will need to be sealed.

There are fabric glues - be sure to get one that is water-proof. Another possibility is to find something called Shoe-Goo. You need to pull the edges of the rip together and overlap them slightly if possible, then glue a new piece of fabric over the edges. Weight it heavily until it dries fully. Make sure that ANY edges or seams are on the outside so that they do not rub the horse.  When you cut the fabric piece, round the corners of it to help them lay flat.

If you want to sew it, then use dental floss as someone else suggested. Bring the edges together on the outside and "whip" stitch it (edges sticking up, then go through all layers, pull floss over the top of the edges, go through all layers again, etc.)  Use a balloon (cut off the neck of the balloon) on your fingers to grip the needle with or use pliers.