I would first determine the fiber content of the fabric and then consult with a professional cleaner. Blood is difficult to remove, especially after it's dried. The longer it's set the more difficult it is to remove and so I'd do this right away. There is a big difference between linen and leather.
You want to know if the fabric will shrink if you use liquids on it. You want to know if spot cleaning is possible.
Another consideration is that if you try to clean it while the fabric is still on the chair you'll be driving the stain into the padding and not actually getting the stain out. This may be OK if the fabric is heavy enough to not reveal the dark color underneath. This is unlikely. And the stain will eventually work it's way back into the fabric.
If the stain is on the cushion you may be able to remove the fabric. Most cushions are encased in a cover with a zipper on the side.
You may be able to more easily remove little droplets with hydrogen peroxide but that depends on the fabric and whether or not the hydrogen peroxide will then leave a "water" mark.
I definitely recommend consulting a professional cleaner.
If the foundation of the chair is in good shape and a slip cover would cost less than a new chair, you can have it slip covered. Depending on the fabric you may not be able to dye it. Man made fibers do not take dye evenly. Some do not take dye at all.
I do wonder why you stayed on the chair with him. I've been working a lot on boundaries over the years. We can take care of ourselves and our children at the same time. It would be rare that a baby would be so injured that you couldn't move him to a better spot. Much of this awareness does come with experience. It helps to widen our awareness after each of these sorts of events. Be gentle with yourself at the same time of being aware that next time you'll let yourself be aware of your surroundings and the consequences of remaining or leaving them.