I would simply stain it or use spray paint. The latex paint is not going to offer any real protection as every time the rug is stepped upon the paint will crack on the fibers and allow moisture to seep into it. At least with a spray coat or stain there will be away for the rug to dry out when it gets wet.
You'd be better off using a semi-transparent stain made for exterior decks and fencing since that'll soak into the sisal plus provide some water resistance. Some mfg's have bases that can be tinted to various colors. You could apply it with a sprayer (hvlp, garden, Wagner hand-held electric buzzer, etc) and do both sides.